In the last fifteen years or so, I have become increasingly disenchanted with the way modern mathematics deals with, or rather doesn’t deal with, the serious logical problems which beset the subject. These difficulties arise from a misunderstanding of the nature of `infinite sets’ and `the continuum’, and then extend further in many directions.
`Infinite sets’ are propped up, according to the standard dogma, by certain axiomatics, which lift the burden of having to actually define properly what we are talking about, and prove the various theorems that we would like to have true. What a joke these ZFC axiomatics are. The entire situation is ironic to the extreme: in fact Cantor’s Set Theory was vigorously opposed by most prominent mathematicians during his day, and then collapsed in a catastrophic heap at the beginning of the 20th century due to the discovery of irrefutable paradoxes. And now, fast forward a hundred years later: not only has Set Theory been resurrected, essentially with no new ideas—most of the key concepts go back to Cantor or Turing, and are just endlessly recycled—but now most of us believe that this befuddled and imprecisely laid out subject is actually the correct foundation for the rest of mathematics! This is little short of incredible. I feel I have woken from a dream, while most of my colleagues are still blissfully dozing.
And our notion of the continuum is currently modelled by the so-called ‘real numbers’, which in fact are far removed from most sensible people’s notions of reality. These phoney real numbers that most of my colleagues pretend to deal with on a daily basis are in fact hazy and undefined creations that frolic and shimmer in a fantasy underworld deep beneath the computational precisions of our computers, ready to alleviate us from the dull chore of striving for precise computations, and incorporating correct error bounds when we can obtain only approximations.
We are talking about irrational numbers here; numbers whose names even lay people are familiar with, such as sqrt(2), and pi, and Euler’s number e.
Supposedly there are myriads of other ones, given by various arcane procedures, formulas and properties. The actual theory and arithmetic of such real numbers is never laid out completely correctly; rather we find brief ‘summaries’ of the wished-for properties that these creatures have, properties that ensure that theoretically many standard computational problems have solutions, even if our computers can in fact not find them.
Ask a modern pure mathematician to make the computation pi+e for you, and see what kind of bemused look you get. Is not the answer the same as the question? Is this not how we all do `real number arithmetic’??
The belief in `real numbers’ supports a false mathematical dream-world where almost everything has a solution; a Polyanna fantasy land which can be conjured up by words but not written down on paper. (Of course the computer scientist or applied mathematician or scientist knows that in reality all meaningful computations occur with rational numbers or floating point decimals).
What a boon it is to live in the `infinitely real’ dreamscape of the modern pure mathematician! To conjure up `constructions’ and ` computations’ these days we need only scribble words, phrases and descriptions together. This is why so many of the ‘best’ journals are filled with page after page of what might be generously called `mathematical prose’. See my submission `Let H be a load of hogwash’ to get a feeling for this language of modern mathematics that the journals encourage.
Most pure mathematicians feel little obligation to address the claims of logical weakness. Objections such as mine may be safely ignored. Unlike scientists, we don’t feel the obligation to step up to the plate and respond rationally to criticism, as it clearly cannot be correct: since the majority rules! As long as we all play along, and ignore the increasingly obvious gaps between what our computers can do and what we are claiming, everyone can pretend that things are merry.
But could the tide be turning? A little while ago, James Franklin and I had a public debate (quite civilized and friendly I would add) in the Pure Maths Seminar in the School of Mathematics and Statistics UNSW, and lo and behold– the room was filled to capacity, people were huddled at the doors from outside trying to hear what was said, and my heresies were not met with a barrage of hoots, tomatoes and derision.
Judging from the many comments, it is no longer such a one-sided debate as it was a few decades ago. I reckon that young people’s comfort and trust in computers has a lot to do with it. What is it really, if you can’t get your computer to model it?? Only a fantasy.
You can join the revolution, too. Don’t be so accepting of everything you are told. Ask for explicit examples and concrete computations. Be suspicious of appeals to authority, or the well worn method of swamping with jargon. And of course, watch as many of my videos as you can, for a slow but steady introduction to: a more sensible world of pure mathematics.
Perhaps the forces of confusion and orthodoxy will soon be on the back foot.
njwildberger: tangential thoughts 
Norman,
In your various videos you introduce algebraic extensions of the rational numbers so as to be able to work with particular irrational values – but only one value at a time. Do you therefore feel that the field of ‘algebraic’ numbers is a fiction too, due to the infinite aspect of specifying every single ‘surd’ which would need to feature in its construction?
The kink in the development of mathematics seems to be the ‘invention’ of the square root: taking any positive rational number (and therefore any two positive integers) we can always perform a division to obtain a valid result. But expecting the results (quotient and divisor) come out as the same number is messing with our permitted concept of divisibility. Fields allow division but there are many, many more ‘square-free’ integers than there are squared ones, so why would we think that root extraction should come as a property of the field?
And as you point out in several places also, the desire to have a commensurable (i.e. rational-valued) hypotenuse on the part of Greek mathematics seem strange when they had discovered that it works every time with squared values! However they had such a powerful (because it was simple) proof for irrationality that I suppose it planted the seed in subsequent centuries that finally led to the accident of ‘real’ numbers. (What a shame)
Thanks for the nice comment. Yes I do not accept the field of all algebraic numbers. We can create extension fields one at a time, but specifying their arithmetic gets harder and harder with each step. We are not in a position to have a field with all quadratic surds, for example.
Ah! From this remark I see, that you basically agree with my comment on how to add algebraic numbers in a mathematical sensible way. You can create extension fields from the rationals. The only thing we disagree on is that if the infinite is banned, both N and Q must also be suspect. At best they can be ideas you can use to construct certain finite sets from.
Hi Konrad, In fact I think we are in agreement that there are no infinite sets denoted N or Z. That’s why I prefer to use the language of type, and refer to a natural number as having type Nat, and a rational number as having type Rat.
In response. I know the concept of ‘type’ from computer languages, which I studied some time ago.
In Delphi for example, to create a button you can click, you must first declare something you want to program, as a certain type. In Delphi you declare for example, a class of type TButton, So what you do is
Button1: TButton;
This only tells the computer that you intend to create a button. But at the moment you declare the type, it does not yet exist. In particular, no memory is allocated to the variable Button1.
Only if you give the following command:
Button1 := TButton.Create
the computer assigns some memory space to the deckaratuin Button1, which only then becomes a variable you can use to provide with data. Only then the button exists in your program, and can appear on the screen. If you try to assign data to a button that is not instantiated, you either get an error message of the compiler, or you get a program that is very unreliable or buggy, because it might use memory that the program uses in another way.
In CBuilder something similar occurs, but it is more visible what happens, because there you base yourself on C, which makes heavy use of pointers. So by the command
Button1^ = TButton^.Create
you reserve some memory for the button code, and you create a pointer which appears as something you can use as a variable in the program you are making.
If you mean by ‘type’ something like TButton, then it is impossible to state:
1 is an element of Int, because if Int is only a TYPE, then the word stands for something that does not exist. So you must then write TInt, which is only a type.
So if TInt is a type, then it cannot contain anything. If you want to have integers you can use to count with, or to do arithmetic with, you must do something like
Int: TInt.
Which still does not mean, that the integer exists.
Only after
Int := SetOfInt[1 … 10.000].Create;
the SET Int exists, of which 5 can be a member.
Only THEN you can make the statement that 5 E Int. But then 5 is an element of the set Int, consisting of 10.000 elements. In other words, a finite construction.
What I am basically saying is that by stating that a natural number has a certain type, you are not precise enough. Are you saying that a natural number HAS a certain type implies, that the natural number first exists, and then is recognized as having a type? But that is impossible, because you must first have the set also, of which the integer is a member. Or are you saying that do you first have the type, and then you create the natural number as an instantiation of that of something else consisting of members all having that type?
In other words: a type does not yet imply existence.
I would say that you are getting at exactly the same thing I tried to communicate with you. Begin with ideas, which you all see as types (I myself use the word ‘principle’ for that, by the way), construct structures that correspond to these ideas, and then apply them. But then our minds are seen as equivalent to computers.
If it works this way, then ‘existence’ must imply ‘instantiation’, which is for the mind the same as construction. Since the mind is finite, it is only capable of constructing finite imaginations. That is the contents of the statement that N, Q and Z cannot exist. And if that is so, then N or Q or Z cannot be constructed as existing sets. At best they can be types. Only sets as instantiations of these types can exist. So if N or Q or Z are types, then they are what I have called previously: ‘potentialities’.
I myself think, that the mind is capable of two things: construction and elimination. Construction leads to experiences, and elimination on these construction based on rules and principles is identical with becoming conscious of these experiences. Both together then lead to what I call ‘conscious experiences’, which are then identical with ‘understandings’. For, in my experience, I only am capable of saying that I ‘understand’ something, if I ‘see it in my mind represented in some form of imagination, as something that is represented as a form of the senses; sound, feeling, or smell. In that way I can truly say, that ‘making sense’ is much like ‘making a sensible representation’ of it. And that implies some form of mental instantiation as explained above.
Well indeed, insofar as a thought can be called conscious, it must be composed of entirely of sensations of the five senses. In principle, then, a perfect explication of what a mathematical object is would tell you directly how to have that experience in your five senses. In practice this would probably always be done with either a static picture or a video.
Then the paradoxical notion of infinite set would be laid bare: one could show a video panning across a tape of points that never seems to end no matter how long you watch (a coherent interpretation of the term “infinity” as merely a process that seems to continue as long as we care to watch), but never a static image that has infinitely many points (the usual incoherent interpretation).
Konrad has hit the nail more squarely on the head: the fact that computers can’t perform infinite calculations or represent infinite sets is convincing…to a point, but there is a more fundamental and ultimately persuasive criticism to be leveled.
It is a breach of the basic norms of communication to speak of “infinite sets.”
As Konrad touched on, the very idea when we utter the term “infinite set” (or any term) is that we have whatever that term refers to in our mind. We have ostensibly already imagined the object in its fullness, and are now making a communicative signal that we hope prompts the listener to reproduce that same object in their mind. In scientific contexts, this imagining must be precise; we cannot pass the buck to someone else, as if we are claiming that “Hilbert could imagine infinite sets in their entirety but I can’t.”
Insofar as communication entails the reproduction of what is in the communicator’s mind, anyone speaking of infinite sets cannot really be said to be communicating. Unless the person can present the notion in a way that allows us to imagine it, they are speaking nonsense. And if they themselves even admit to being unable to imagine an infinite set, they admit in the same breath to be speaking incoherently. They’ve just told us that they literally don’t know what they’re talking about.
You wrote: “Insofar as communication entails the reproduction of what is in the communicator’s mind, anyone speaking of infinite sets cannot really be said to be communicating. Unless the person can present the notion in a way that allows us to imagine it, they are speaking nonsense.”
Alas, it is more subtle than this. Our minds are capable of two functions, not one. We are not only capable of imagination, but also of elimination. This difference lies at the heart of the debate between Brouwer and Hilbert, of which Norman Wildberger is one member, having chosen the side of Brouwer alone.
Thoughts do not represent consciousness, but represent experiences. The five senses provide us with experiences. In the East they talked about 6 senses, not 5. The ‘6th sense’ also was called ‘the indirect sense’.
To understand what they were referring to, I need to make a little digression.
Modern brain scanners make it somewhat possible to see how information flows through the brains. When they looked at sleeping brains, they expected that the brains would not be active. And for large spans of time when people sleep, this is what happened.
But to their surprise they also saw when people sleep, and dream, the brains are even MORE ACTIVE than when awake.
Another observation was, that whey they looked at brain activity for the first time, they EXPECTED to see information flowing from the senses. But what they SAW, much to their surprise, was not ONE, but TWO information flows. One originating from the senses, and a second one originating from the frontal lobes. Apparently the first information flow from the senses operates on the information flow from the frontal lobes. What it does is EXTINGUISHING most of this information flow, until only a little is left. This little, that is left, is then ‘the experience of the outside world’. It is exactly what most people mean when they speak of ‘conscious experience’.
This is also consistent with the finding of the enlarged activity of the brains in dream sleep. When we sleep, the information coming from the senses is blocked. Therefore, when we dream, the brains generate the same information as when we are awake. But now none of this information is operated on by information of the senses. Therefore NOTHING is eliminated. The result is that no thought is inhibited in any way. The result is brain activity.
Another fact is the following. If you go to your home, and you open the door, what you THINK is happening is that you SEE the door. That is, information enters your senses, causing a picture in your mind on which you act.
But that is not what happens. What happens is this. The brain tries to generate the thoughts that represent all worlds you can imagine. But most of these possible worlds do not correspond to the actual world, and even to the actual situation you are in.
That is where the senses come in. The data of the senses operate on this process of generation of all possible worlds, and then eliminate most of them, until only the one that is NOT INCONSISTENT with the data of the senses remains. And THAT then represents the experience of ‘the world’ and ‘the actual situation you are in’.
To read more about this, I refer to two books: ‘On Intelligence’ from Jeff Hawkins, and also to the book: ‘i of the vortex’ from Rodolfo R. Llinás. The first book makes clear, that the senses are way too slow to create a picture of the front door from the senses. And the second book explains, why, when we are asleep, our experiences are just ‘as real’ as when we are awake. Or, to be more precise, when we are asleep, and dreaming, we do not know that we are sleeping and dreaming. In other words, there is no fundamental difference between the experiences in the dream state and the experiences of when we are awake.
Nevertheless, there IS an asymmetry between dreaming and being awake. When we are asleep, we do not know that we are asleep. But when we are awake we DO KNOW that we are awake. Moreover, at the time we are awake, and when we remember our dreams, we RECOGNIZE that the experiences we had in our dreams were ‘not real’.
Based on all this, I claim the following. Brain activity represents EXPERIENCE. But CONSCIOUSNESS, is the exact opposite. It represents INHIBITION, resulting in ELIMINATION.
So the difference between being awake and dreaming is that when we are dreaming, we are experiencing, but we are NOT conscious. But when we are awake we are BOTH experiencing AND we are conscious. So the cause of the asymmetry between dream-experiences and the experiences when we are awake, is consciousness, and its eliminating, and inhibitive functioning.
Moreover, so I claim, there is a fundamental difference between experience and consciousness. Experiences are formed by thoughts. But consciousness is the result of a process of ELIMINATION of thoughts.
This brings me to the remark I made about the East. They speak of a 6th sense. This 6th sense is indirect. I think this 6th sense is nothing else than a name for consciousness itself!
Moreover, they have made a discovery. It is possible to train the mind in such a way, that ALL thoughts are eliminated. The result is ‘consciousness without experiences’, that is ‘consciousness without experiencing anything’. This state has been discovered several times in the East. Notably by Lao Tzu and by Buddha. It has several names. In India, Buddha called it Dhyana. When it was introduced in China, the name changed into ‘Chan’. And when it was introduced in Japan, the name for this state became Zen. Within Tao it is just called Tao. Here, in the West, it is loosely called ‘enlightenment’, although almost no awareness exists about what it really means.
In any case, Zen Masters discovered several ways to bring yourself into this state. The most effective method they found was the Koan meditation; you take a statement, that is constructed deliberately in such a way, that NO picture can EVER satisfy it. And when you meditate on it, it might happen that you succeed in UNDERSTANDING, that, indeed there can be NOTHING that can POSSIBLY correspond to the statement. ‘What is the sound of one clapping hand?’ is one of the most famous examples of such a Koan.
And, to return to your statement, the assignment a Master gives to meditate on a Koan is an attempt of communicating ‘nothing at all’. You ONLY understand the Koan, if you REALLY UNDERSTAND, that there is NOTHING that CAN POSSIBLY satisfy the message of the communication.
So, it is possible to experience without being conscious. This is what happens when you are asleep, and when you are dreaming. It is also possible to be fully conscious, without experiencing anything at all. This is the ‘state of Dhyana, Chan, or Zen’, which is sometimes also called: ‘No Mind’, and which is for most people, beyond what they are capable of with their minds.
And this brings me back to mathematics. Corresponding to both experience and consciousness, there are two types of mathematics. You have, let me call it ‘constructive’ or ‘analytic’ mathematics. And you have ‘synthetic’, or ‘axiomatic’ mathematics. When Hilbert and Brouwer had their debate, (which, by the way, was never resolved. People could not solve it then, so they just stopped it, realizing that both had a point) they were talking about different things. Brouwer was talking about the construction of thoughts, while Hilbert was talking about the elimination of thoughts. Moreover, there was misunderstanding about logic itself. Russell and Whitehead believed, that logic could be a tool for the construction of mathematics. Russell even believed you could derive all of mathematics from logic alone. You very likely know, that Kurt Gödel proved them wrong. Logic is a tool with which you can test two things; completeness and consistency. In any case, logic is a tool for testing the soundness of mathematical theories. (Soundness in the logical sense.) The purpose of testing is, as Popper has clearly shown, not the discovery of new truths, but the elimination of falsehoods. And therefore logic itself is a tool for elimination, not for construction.
So the question: ‘is mathematics a branch of logic, or is logic a branch of mathematics?’ is a false question. The relationship between logic and mathematics is the same as that of any theory of physics and the experimental test. You cannot say that physics is a branch of the theory of experimental testing, (because there is no such theory apart from the theories of physics you want to test) and you can also not say that experimental testing is a branch of physics (because there is no ‘law’ of experimental testing). No, the relation between physics and the experimental test is epistemological. The fact that we have a way to check whether our theories of physics are sound, namely through the experimental test, MAKES that physics to be a science. In the same manner, the fact that we have a way to check whether our mathematical theories are sound, through logical testing MAKES that mathematics is a science. Any body of knowledge, that claims to be a science, but lacks such a test, is not a science. (Think about psychology, sociology, or economics.)
The conflict between Brouwer and Hilbert can be resolved if you recognize all this. It makes us see that they were talking about two aspects of mathematics, both involving two different activities. You have CONSTRUCTION and you have ELIMINATION. Both lead to different forms of mathematics. One leads to analytic mathematics, and the other to synthetic mathematics.
When you are dealing with mathematics through algorithms, you are basically acting in a constructive way. If you are dealing with mathematics through axioms, you are basically acting in an eliminative way. If you construct an axiomatic system, you are stating rules certain mathematical constructs have to satisfy. And then you are basically acting in an eliminative way. This is because, though stating axioms, you only want to consider everything satisfying those axioms, without making the structures that satisfy those axioms explicit. And THAT means, that you are EXPLICITLY stating what you temporarily REJECT. And that is eliminative.
This is also where the so-called universality of the axiomatic method comes from. By not making the structures that satisfy your axioms specific, you imply that there can be many structures that satisfy those axioms, without making any of them specific. And that is exactly what universality means.
Computers, on the other hand, DEMAND that you make the structures explicit. Therefore if you fuse computers with mathematics, you become more interested in analytic methods than in synthetic methods.
This digression is enough to resolve the conflict between Brouwer and Hilbert. It revolves around ‘The Law of Excluded Middle’: Does the proof that a contradiction follows from a certain assumption shows that the denial of the contradiction ‘exists’? Hilbert claimed that it did, and Brouwer claimed that it did not.
In other words, if for example, you prove that the assumption that there are a finite number of primes leads to a contradiction, you have proved that it is not true that there are only a finite number of primes. Hilbert would interpret this result as a proof that the SET of primes is infinite. But BROUWER would interpret this ONLY as a proof that ANY set of primes MUST BE finite, otherwise it does not exist, BUT it can be extended. Therefore ‘not finite’ does not mean ‘actually infinite’ but only means: ‘it is possible to ENLARGE any finite set of primes, through adding another prime to that set not in that set, without leading to a contradiction’. Therefore we are talking not about actual existing sets, but about sets that have an extra attribute: the attribute of extendability, that is, the potential of extending them without causing a conflict between the new set and the definition that was satisfied by to the original set, and which might have defined it in the first place.
To say it more bluntly; Brouwer talked about the EXPERIENCE of mathematics, and Hilbert talked about the CONSCIOUSNESS of mathematics.
Communication does not only consist of imaginations, but also of statements telling us what imaginations within a certain context to eliminate, and what is required for eliminations to be possible.
To give a simple example; take the concept of ‘even number’. How can we define this exactly? You can say: a number is even if it is divisible by 2. But all rational numbers are divisible by two, so that is not precise enough. The set on which to operate the division, the context, must also be made explicit.
So, to make sense, you must first tell the context. That is, the set within which the concept of even number makes sense. This might be Z. And then you can say something like:
E ={ x € Z | x/2 € Z }. That is, those elements of Z such that if you divide that element in Z by 2, you get another element that is also in Z. What you then have communicated is that 4 is even, but 5, for example, is not, and therefore should be ELIMINATED from all possible candidates of Z that might be placed in E.
And to be REALLY precise, you also should introduce a largest number, defining Z, so that is clear from which EXISTING set you are taking the elements to be placed in E.
This means, that Z is not ONE set, but just a name of a COLLECTION of POSSIBLE sets within which you make the elimination. And that shows, that the definition of Z belongs to the realm of synthetic mathematics, and not to the realm of analytic mathematics.
Z then becomes a name for large number of different existing sets satisfying the definition. Something Norman Wildberger calls a Type.
To conclude, I want to add a remark. Norman Wildberger’s misgivings come for a large part from computers, and what they tell us about mathematics. Although the computer is a very powerful tool, it is only a tool for CONSTRUCTION. Therefore it is an aid to our EXPERIENCES. A computer is, in essence, not a tool for elimination. Therefore it can only shed light on analytic mathematics, not on synthetic mathematics.
But that is how it is at present.
In laboratories people are busy to construct a new kind of computer. The quantum computer. This computer works in a fundamentally different way. In essence, the quantum computer makes use of the fact, that in quantum mechanics you can have destructive interference. That is, ELIMINATION. Even elimination in a Turing like way, that is, UNIVERSAL elimination, in the same sense that our present computers are representations of UNIVERSAL generation. (See Steven Wolfram’s book: ‘A New Kind of Science’ for more about this.) This means, that this type of computer represents CONSCIOUSNESS. I therefore predict that the quantum computer will do for synthetic mathematics what the present computer does for analytic mathematics.
A future computer that incorporates both the capacity of the normal computer and the quantum computer would be one that has the same relationship with the world we have. It would be a computer capable of conscious experiences, just like we do.
Konrad,
Thank you for the fascinating reply about imagination and elimination, which correspond to explicit representation and “x” – that is, representation of the general, or partial information. In some sense I doubt it is a proper definition to specify a set by forcing the listener to solve a puzzle or do a proof to figure out which elements are in the set and which are not. While often this would be all right, it is a type of definition that could be easily abused by forcing the listener to do very difficult or impossible things just to understand what the mathematician is talking about.
That aside, my point in saying, “Unless the person can present the notion in a way that allows us to imagine it, they are speaking nonsense,” was not just about explicit imaginings. Perhaps I should have said “allows us to experience it” instead. What I am saying is that even our understanding of “x” or of “some prime p” is ultimately an experience the speaker ostensibly has, of some sort. Elimination is also part of that experience. Whether it is truly possible to represent generalizations in one’s mind, and however that might be accomplished, it ultimately must have happened in the speaker’s mind for the words to have any referent in the communication.
In the case of the term “infinite set,” it is not a matter of analytic vs. synthetic, but simply a matter of equivocation on the term, at best, or complete incoherence at worst. If the mathematician justifies to herself the use of the term “infinite” by imagining a clear video of an process that continues seemingly without end, then turns around and uses this fundamentally DYNAMIC term to describe a STATIC concept (a set; represented by an still image rather than a video), that is an equivocation. She is failing to communicate even with her own self.
That is the point I wanted to get across. Because once properly understood, this should convince just about everyone.
—
Getting back to the interesting question of how to represent non-explicit objects, or objects about which we only have partial information based on some eliminative characteristic (“x is a natural number,” “x is even,” “x is prime,” etc.), I was interested to see the progress made in diagrammatic approaches to mathematical and logical foundations because researchers there have run into the problem of how to represent partial information in diagrams. For example, an Euler diagram allows you to simply read a logical conclusion off the diagram, as reading a map, so it is potentially very powerful, and also being completely constructivist it enables no contradictions. That is, since all the logical inferences of the starting assumptions are baked into the diagram and can be read off it, contradictions are automatically impossible to represent. This seems like a good thing and part of what Brouwer was aiming for with his approach. (See http://plato.stanford.edu/entries/diagrams/)
I have been working on expanding the expressive power of Euler diagrams myself, and representing such partial or information is particularly difficult because of how explicit diagrams are. However, blacking out part of the diagram can work in certain cases, and in others I simply ask myself how I do it in my own mind. What am I seeing when I think of “x” or of some unknown even number? What I have found is that insofar as mathematical situations can be completely represented diagrammatically, many the results we would usually have to prove simply pop out from the diagram at us. That suggests some possible exciting new approaches that might even work on some heretofore intractable problems.
In any case the discipline introduced by having to actually spell out for the audience how to visualize a mathematical object or situation (even if that entails some kind of generality it must somehow be able to be thought of, or else how can we speak of it!?) seems in line with Norm’s approach of being strict about rigor even when it makes things much harder, because the result is more honest, more interesting and rich, and ultimately – if successful – more applicable to real world problems.
Thank you very much for the link!
Also, very interesting what you are working on. I have also been thinking for quite a long time about the connection between abstract thinking and visual representations.
There is one book I found particular helpful. It is the book: ‘Geometry’ of Brannan et al.
http://www.cambridge.org/nl/academic/subjects/mathematics/geometry-and-topology/geometry-2nd-edition
One of the things it made clear to me, was that the understanding of abstractions is not only a matter of visuals. A real understanding of abstractions happens through the use of both symbols and visuals.
To be specific, take the concept of a triangle. The book says that this can be done by considering just ONE triangle, and then attach to it the entire group of affine transformations in the plane. This group is able to transform this one triangle into all possible triangles. So the concept of ‘triangle’ consits of (Triangle, G(Affine Transformations)). In the same manner you can define parabolas, hyperbolas and ellipses. An affine transformation can transform an ellipse into an ellipse, a parabola into a parabola, and a hyperbola into a hyperbola, but NOT a parabola into an ellipse or a hyperbola etc. So, within the context of affine transformations, the concept of parabola, hyperbola and ellipse are distinct.
If you take a more general group, the group of projective transformations, and you apply this to a parabola, it IS possible to transform it into an ellipse etc. This then leads to the CONCEPT of ‘quadratic form’, consisting of just ONE quadratic form, say, an ellipse, and then you can transform it into a hyperbola etc. In this way you can see that the concept of quadratic form is more general than that of an ellipse etc, and you make exactly clear in what sense.
Moreover, you can transform a triangle into two parallel lines intersected by a third. The well – known property that the sum of the inner angles of a triangle adds to 180 degrees can be recognized as the theorem that if two lines are intersected by a third, then the sum of the two inner angles on the same side of the intersecting line adds up to 180 degrees. This is because the magnitude of the third angle is, if you consider this as a triangle, zero.
I shall later return to your response, because I am now a little pressed in time.
Thanks!
Dear Ishinden. You wrote: “Perhaps I should have said “allows us to experience it” instead. What I am saying is that even our understanding of “x” or of “some prime p” is ultimately an experience the speaker ostensibly has, of some sort. Elimination is also part of that experience. ”
I disagree. Elimination is part of understanding. But not all understanding is experience. Moreover, elimination is neither a part of experience, nor is it something that can be experienced. Eliminations cannot be experienced, but they CAN be UNDERSTOOD.
To give a simple example: suppose I give you a closed box, and say say that there is NO cat in that box. (Assuming that there is, indeed, no cat in that box.) You open the box, and see that, indeed, it does not contain a cat.
Now what have you experienced? Have you experienced the cat? Obviously not, because there was no cat in that box. Have you EXPERIENCED that there is NO cat in that box? But that is absurd! How can you experience the ABSENCE of something?
No. What happens here is something else. You have UNDERSTOOD what I have said,and to test my statement you open the box, and compare it with the image in your head of a cat in a box, and see that what you experience is IN CONTRADICTION WITH the picture of experiencing seeing a cat in that box. In other words, you EXPECTED you would NOT experience the presence of a cat in that box, of which experience you had a clear picture. And then you found that, indeed, the experience of finding the cat in that box was not occurring, just as expected.
Or, to say it more simple, the LACK OF experiencing the cat in the box made you realize,that what I had told about the context of the box was correct. And with ‘correct’ I mean, ‘representing the understanding of the statement about the box’.
The mistake you made is a deep one. Maybe you can find in one of my commentaries, that I deny the existence of the empty set, exactly on these grounds. I understand what is meant by the empty set. But there is no such thing like the empty set. Assuming otherwise leads to the contradiction of stating, that non-existence is a form of existence, instead of the absence of it.
The existence of the symbol of the empty set proves not that the empty set exists, but that we have a symbol for the UNDERSTANDING that there are mathematical statements that do not lead to experiences.
In fact, it was thinking about, and meditating on the meaning of the empty set, what made me succeed for the first time, to stop all thinking, and to discover the brain state whereby there is no experiencing at all, but nevertheless UNDERSTANDING of what was happening. To be exact, I in a deep meditation on a Koan I tried to understand it, and then there came a moment whereby I UNDERSTOOD that there could be NOTHING that could correspond to the MEANING of the Koan. And THEN I understood, that THAT WAS the MEANING of the Koan.
Before that, I lived in the confusion of believing that there ‘is’ something like ‘the empty set’, because I thought that it could be experienced. But, again, it is the LACK OF ANY experiencing, but nevertheless UNDERSTANDING that this is what was happening, which is the contents of what Zen is all about. It allows me to understand the very distinction between experience and consciousness. Or, to be more exact, it ’causes’ a mutation of the brains, whereby experience and consciousness are UNDERSTOOD to be completely distinct, and therefore to SEPARATE from each other.
You also wrote: “If the mathematician justifies to herself the use of the term “infinite” by imagining a clear video of an process that continues seemingly without end, then turns around and uses this fundamentally DYNAMIC term to describe a STATIC concept (a set; represented by an still image rather than a video), that is an equivocation. She is failing to communicate even with her own self.”
My statements about infinite sets do not correspond exactly to this, thanks to the fact that I make a distinction between experiencing and consciousness. Again, an understanding does not have to be an experience. It can also mean that you UNDERSTAND an ABSENCE, which is the LACK OF ANY EXPERIENCE.
In the case of the infinite set something like the cat – example above happens. The proof of the infiniteness of any set happens in the following way. You ASSUME that the set is finite. Then, through some logical analysis you derive a CONTRADICTION from that assumption. And therefore you UNDERSTAND that the set you are investigating does NOT have the property that it is finite. In most cases this means that you have proved, that there is no largest, or last member. So, again, it is the ABSENCE of something you UNDERSTAND you express with the word: ‘infinite’.
The mistake most mathematicians make, is that they follow Cantor, and then still take this as that there is ONE set that is infinite. But the absence of a property does not mean that ‘the absence of a property’ CONSISTS OF ‘the property of an absence’.
THAT is what I think is what the confusion is about. They prove the absence of a property, and then take this as ‘the property of an absence’.
If you see though this misunderstanding, then there is only one way out. If you prove that from the finiteness of a certain well-defined set follows a contradiction from its very definition, then the only thing you can conclude from that is that you can EXTEND that set WITHOUT violating the property that the set defines.
In other words, if you prove that the set {2, 3, 5, 7, 11, 13 } consists of prime numbers, and that it is ALWAYS possible to enlarge that set by adding 17 to it, or 19 to the thus extended set etc.,, without the property that the set only consists of prime numbers will be lost, then from this it follows that you can construct, from the property of prime – number, as many sets you like consisting all of prime numbers. So does not prove the existence of ONE INFINITE set, but of MANY FINITE sets, all distinct from each other (because they differ in the existence of at least one element), all consisting of elements of the same kind.
Moreover, if you do not take a specific largest element in a set, but just call the largest element of such a set the n-th element, then what you have proved is that there is no upper bound to the number of sets you can construct corresponding to the property defining the set.
Another way you can express this, and that is indeed how I do this, is that all such sets are POTENTIALLY EXTENSIBLE. So I use the term ‘potentiality’ to designate such sets.
You might dismiss much of the above as just a matter of definition. In particular of how you define ‘ experience’, and how I define it. But that would be a mistake. Definitions are the RESULT of understandings, and not the other way around. And that means, that there is a real understanding to be gained by the way I define things, compared with how you do this.
Hello again, Konrad.
I don’t think we are necessarily in disagreement on the main point, just that words make it difficult to specify exactly. Hopefully I can make that clear here.
“You have UNDERSTOOD what I have said,and to test my statement you open the box, and compare it with the image in your head of a cat in a box, and see that what you experience is IN CONTRADICTION WITH the picture of experiencing seeing a cat in that box. In other words, you EXPECTED you would NOT experience the presence of a cat in that box, of which experience you had a clear picture. And then you found that, indeed, the experience of finding the cat in that box was not occurring, just as expected.”
Here I am calling the finding of what is actually inside the box being in contradiction with the picture in one’s mind as itself an experience. That is, some kind of sensory movie, including the surprise or confirmation of the expectation. Without some deep introspection we can only guess at the specific sensory aspects, but for example the first few frames of the movie in your head might be an image of the box in a cutaway style image with the cat inside but the cat being translucent or maybe having a red X over it, or maybe flashing in and out of view. Then the box opens and it appears empty, and the previous presentation (translucent, red X, or flashing) is replaced momentarily by a few frames of empty box, perhaps followed by emotional sensations of surprise felt in one’s chest or stomach, perhaps then a brief flash of the translucent/X-ed/flashing cat becoming more translucent or having a bigger and redder X or flashing more urgently, followed by frames of just the empty box.
Again, who knows how it actually happens in terms of raw sensations in a particular person’s, but my point is that it happens it some way. And the larger point is that it we should expect anyone putting forth a definition of a mathematical object to ensure that, first of all, they themselves have experienced some consistent set of sensations that they are referring to by the term, and second of all to define the object in a way that allows the audience to do so as well.
I don’t wish to take anything away the idea that there is a meditative state where we can experience a kind of zen nothingness, but I do think that there is an important difference between that state and merely the experience of absence of a particular thing one was expecting. The first is literally no experience, agreed, but the second is an experience of actual disappointment/confirmation/surprise/etc. and preceded by certain concrete images.
A more fundamental way of considering how we understand absence of a cat in a box is to take the question back to utility – that is, back to, “Why does it matter to me?” In my case, if I had any reason to believe there was a cat in a box nearby I’d suddenly want to open it. If someone then told me there was no cat in that box, I’d probably be less likely to open it. The very meaning of “there’s no cat in the box” to me would resolve to specific utility-function guideposts saying things like, “LOOK ELSEWHERE TO FIND A CAT.” (I’d see a movie of opening the box and being disappointed, or maybe a movie of bypassing the box and finding a cat somewhere else.)
So ultimately I think the question of how we model/understand/experience the absence of particular objects is just another way of asking how we do the same for generalities. In this case the generality is “every situation that doesn’t involve a cat in the box.” I am in basic agreement with Hume in that generalities are modeled in our minds as bundles of specifics. For example, “Look elsewhere to find a cat” or a movie representing something like this would be one such specific. In that way, elimination or absence (of a particular object) as you are talking about would just become a collection of specific experiences. Again, zen-like absence of any experience at all is something different from this more specific absence.
Regarding the empty set, it’s unfortunately complicated by the fact that “set” has no clear definition. Under a typical definition that a set is “a collection of objects,” the term “empty set” becomes impossible to interpret. (Like a koan, yes, but then again every koan resolves to the same thing – no experience – so we can’t really say we are understanding the term “empty set” in a way that was ostensibly intended by the mathematicians who uttered it, since it means the same thing as “one hand clapping”; we should hence be able to replace all instances of “empty set” with “one hand clapping” without harm to such mathematician’s theories.) Or certainly we can say that there is little expectation that everyone would understand the same thing by the term, which would disqualify it as a term of science.
More generally, terms like zero and space are all kind of placeholders rather than actual objects. They basically say, “There is no number or physical object here.” (In terms of concrete experience, it would be for example, “See a movie of yourself being disappointed if you go looking for a number/object here.”)
I fully agree on the idea of infinite/finite in that if you prove that a set (or perhaps a type of set) is not finite, it simply means that it can be expanded arbitrarily without breaking the qualifying properties for inclusion in the set. The more careful mathematicians seem to take some pains to speak of, say, a sequence increasing “without bound” rather than “to infinity.” Since I think infinite is most usefully thought of as an inherently dynamic concept, concerned with incessant motion, it seems cleaner to say that sets can be expanded (verb) for as long as we care to bother, rather than infinite (adjective), or perhaps just that there are arbitrarily large sets of, say, primes that one could create.
With regard to triangles and other geometric forms, here again we have a situation where in this particular case (reminiscent of what Norm calls “baby examples”) we have a definition that, even though it forces the reader to perform a test for each object to determine if it falls into the category that the mathematician is talking about, the test happens to be so easy that it’s not a problem. However, the practice of defining terms by forcing the audience to solve a puzzle to figure out what the speaker is talking about is dubious in general. Consider the Collatz Conjecture (the 3n+1 problem) baked into a sequence, as mentioned in a recent Math Foundations video, where the audience has to prove or disprove a longstanding unresolved conjecture in order to figure out what the mathematical object really is, or what its real definition even is. We could similarly define a “Collatz set” as the set {0} if the Collatz conjecture is false and {1} if it is true.
So-called functional definitions are not in general sound, even if there are some examples where they seem to work out fine. It seems to me a mathematician should always be careful to distinguish between an explicit definition and one that requires the reader to jump through hoops (perhaps impossibly) in order to make the definition explicit.
You wrote: “Here I am calling the finding of what is actually inside the box being in contradiction with the picture in one’s mind as itself an experience. ”
The matter is not that easy resolved. Let me point out that there WAS no contradiction between what was in the mind and in the box. I wrote: “In other words, you EXPECTED you would NOT experience the presence of a cat in that box, of which experience you had a clear picture. And then you found that, indeed, the experience of finding the cat in that box was not occurring, just as expected.”
Moreover, by saying: ‘some kind of sensory movie’ it is clear to me, that yo do not really see what happens in the mind when you did not expect to find a cat, and you indeed do not find a cat.
The point is: you cannot think about: “NOT a cat” without having a picture of the cat. There is even a famous Zen story that makes this point that goes a little bit like this. “A student came by a Master, and asked about how to become enlightened. The master said: it is easy. Just succeed in not thinking about a cat for ten minutes.” The student went home, very happy. After all, in his ordinary state he was not thinking aout cats anyways. But then, to his surprise, whatever he tried, he could not stop himself thinking about cats.” End of story.
You write: “Again, who knows how it actually happens in terms of raw sensations in a particular person’s, but my point is that it happens it some way. ”
And that is where we differ. What I claim is, that when you understand the meaning of the sentence: ‘no cat in the box’, then what is happening is a picture of a cat in the box, and then an elimination of that cat, so that what is left is a picture of a box with no cat in it, and then understanding that THAT picture is a representation of the sentence: ‘no cat in the box’ much like how the concept of triangle in affine geometry consists of just a SINGLE triangle and an abstract matrix [[a, b], [c, d]] which is not singular, whereby you can choose for a, b, c, and d any four magnitudes. So just like a ‘general triangle’ consists of (Tr, Affine Transformation), a box without a cat in it consists of (Box with cat, Elimination transformation).
In other words,within mathematical understandings there is an interplay between symbols and pictures. And what I claim is that the pictures are the experiences, and the symbols represent the eliminations. I even dare to claim that ANY understanding consists of pictures + eliminations. The pictures then are the part corresponding to experiences, and the symbols are essentially operators of elimination, and therefore representations of processes of elimination.
Seen in that light I consider propositional logic, together with Venn diagram (patterns of shading and not shading) as the most basic description of understanding as such. The propositional sentences are then the operations of elimination, resulting in parts not shaded. And the pictures of Venn diagrams are then what is operated on, so they represent experiences in their most abstract form. Understanding a statement like: p & q therefore consists of the whole diagram, wherein everything is eliminated, except that what corresponds to both p and q.
You wrote: “I don’t wish to take anything away the idea that there is a meditative state where we can experience a kind of zen nothingness, but I do think that there is an important difference between that state and merely the experience of absence of a particular thing one was expecting. The first is literally no experience, agreed, but the second is an experience of actual disappointment/confirmation/surprise/etc. and preceded by certain concrete images.”
Here you express what you understand, and from it I see that you do not see clearly what happens in your mind, although you see it partly. Indeed, there are preceding concrete images. Those are the experiences. And then there are eliminations of those concrete images, which eliminate not only the images, but also the experience of the images. And then a discovery is made. If all images are gone, then all experiencING is also gone! This allows the Zen master to identify experiencing as such as the psychological manifestation, the ‘feeling’ if you like, of ‘that there is existence’, of ‘is-ness’, and therefore of experience itself. Just like there is no speaker if there is no speech, (there is no ‘I’ that speaks, apart from the speaking itself), there is no experiencing when there are no imaginations. Therefore, the imaginations themselves are the ‘carriers of experience itself’. Or, ‘experience and imagination are the same thing’.
After this elimination nothing is left, except the capability of describing what has happened, which is a description in language, and therefore a translation of the elimination into symbols. And the fact that you CAN translate this elimination into such a description, so I claim, is exactly what is the essence of CONSCIOUSNESS. Moreover, this ability causes itself awareness. (not experience, but awareness), the awareness of ‘that there is nothing that is left’, without, (which is the hard part) SOMEBODY who is aware of this. In that moment it is therefore also understood, that the ‘I’ itself is a product of imaginations. Or, better, the result of the thoughts consisting of speech. So the speaking is the speaker. Apart from speaking, which is understanding itself, there is no ‘I’ WHO understands. This is also the reason, why Zen solves what I call: ‘the religion problem’, the reason why religions exists. This is the problem of our observation of our own finiteness in time. That is, the clear awareness that one day we will die.
If the ‘I’ itself can be made to disappear in full consciousness, then ‘the center of the fear of death’ can be made to disappear. And therefore the Zen state is itself an answer to the problem of our own finiteness.
But I am wandering from the subject.
Let me give another example to try to make the distinction between experience and consciousness clear. One I give many times in my courses on logic. Suppose you ask somebody whether he knows what a cat is. For that person, it is not enough to point at a cat and tell me that THAT is a cat. This is because in the next moment he can point at a dog, and even a door, and claim that those are also cats. No, he must also be able to point to ANYTHING that is NOT a cat, and TELL that what he is pointing at, is NOT a cat. Only if he shows that he is capable of doing this last thing, he demonstrates that he UNDERSTANDS the difference between cats and other animals and objects. And only THEN he is able to convince fully that he knows what a cat is.
From this I claim, that EVERY understanding consists of TWO parts. A part consisting of PICTURES, and a part consisting of ELIMINATIONS. The PICTURE part, so i claim, corresponds to experiences. And the ELIMINATION part, so I claim, is what is the real meaning of the term ‘consciousness’.
TOGETHER they form ‘conscious experience’. In other words, if you look at a cat, and UNDERSTAND that it is a cat, then this understanding consists not only of the picture of the cat, but also of an awareness of that there are animals and things that are not cats. This means, that conscious experience consists of two parts, which also what people REALLY mean with the term: ‘conscious experience’. And that is what I try to make clear to you. When you talk about experience, you are talking not about just experience in my eyes, but about conscious experience. You demonstrate that you do not recognize that the word ‘experience’ stands for two things, not one.
This is also what I see you do in your descriptions. When you use the term: ‘experiences’, you are really meaning ‘conscious experiences’, which are UNDERSTANDINGS. And if you use the term: ‘experience’, you are really talking about CONSCIOUS experience, and therefore about understanding as such.
However, you fail to see that in any conscious experience there is an element of elimination. Therefore you use the terms ‘consciousness’ and ‘experience’ interchangeably.
The remark about utility is further from the mark. This subject is very vast. In fact, the whole of the so-called Austrian economy, is one huge theory based on utility, from which they try to derive all of economy. But even then, it has nothing to do with what we are discussing here: to understand what happens in the mind when we use symbols and pictures. This is because the question what happens in the mind when we use symbols and pictures is independent of intent, and therefore independent of what Ludwig von Mises calls: ‘action = purposeful behavior’, in his book: ‘Human Action, and what Rothbard describes as ‘motive’ in his book: ‘Man, Economy and State’.
So talking about this is definitely NOT ‘more fundamental’. On the contrary, if understanding in general consists of both pictures AND eliminations, then that whole theory can only be a special case of what we are talking about, and therefore is entirely beside the point.
You wrote: “So ultimately I think the question of how we model/understand/experience the absence of particular objects is just another way of asking how we do the same for generalities. In this case the generality is “every situation that doesn’t involve a cat in the box.” I am in basic agreement with Hume in that generalities are modeled in our minds as bundles of specifics. For example, “Look elsewhere to find a cat” or a movie representing something like this would be one such specific. In that way, elimination or absence (of a particular object) as you are talking about would just become a collection of specific experiences.”
Yes. I already had picked that up about you. The fact, that you understand generalities as omissions. But seeing it that way is exactly what I disagree with. It is close, but not yet exactly it. Understanding generalities does not consist of ommisSIONS, but of THE VERY PROCESS OF MAKING OMISSIONS. That is, THE VERY PROCESS OF ELIMINATION, and not the omissions themselves. It is a mistake much like one you do not make: thinking that the absence of a property is the same as the property of an absence. You are aware that in one way or another omissions play a part in universals, but fail to see that it is one step more abstract. OmisSIONS are not what makes a universal, but omisSION is what makes a universal.
Let me be more specific. Our neurons in our brains perform two tasks. They generate electrical impulses, and they inhibit them. When both processes occur, there is ‘conscious experience’. However, it is possible to have a brain state wherein there is NO elimination. This is exactly what the dreamstate is. The part of the brains that eliminates is then switched off. This is the state, wherein there are a lot of experiences, but no understandings. This is also the state wherein you experience, but are not conscious. So when you sleep, and dream, you are experiencing. In fact, you DO NOT KNOW THAT you are sleeping, because there is NOTHING that is eliminating, and therefore nothing that is able to make the distinction between the experiences of the dreamstate and the experiences that happen when you are awake. You experience dreaming as if you are in another, yet familiar world. The thing that distinguishes the dreamstate from the state of being awake is that there is NO UNDERSTANDING. There is ONLY experiencing.
Only when you are awake, the eliminative activity of the brains, what I have identified as consciousness, is present. Therefore, when you are awake, and you happen to remember some of the experiences of your dream state, you can recognize them as illogical, and therefore as experiences that are not real.
In other words, it is the eliminative part of our brains that causes us to designate certain experiences as real and others as not real. If that is absent, ALL our experiences are completely real to us. And that is why, when we are sleeping and dreaming, we do not know that we are sleeping and dreaming.
What I see you do, is ‘sticking to Hume’s paradigm of generalities’. That is why you fail to see what I try to communicate to you, despite your great intelligence (of which I am also very aware.)
You wrote: “Regarding the empty set, it’s unfortunately complicated by the fact that “set” has no clear definition. Under a typical definition that a set is “a collection of objects,” the term “empty set” becomes impossible to interpret. (Like a koan, yes, but then again every koan resolves to the same thing – no experience – so we can’t really say we are understanding the term “empty set” in a way that was ostensibly intended by the mathematicians who uttered it, since it means the same thing as “one hand clapping” we should hence be able to replace all instances of “empty set” with “one hand clapping” without harm to such mathematician’s theories.) Or certainly we can say that there is little expectation that everyone would understand the same thing by the term, which would disqualify it as a term of science.”
Not quite. You can UNDERSTAND that there is nothing to understand, simply because the same mental activity that allows us to understand, consists of elimination itself. So what is left is not just ’emptiness’, but the very process of elimination itself, which, I claim, IS consciousness, and is present in all forms of understanding. That is why it is possible to understand a Koan. And that is also how you can understand the term: ’empty set’.
So you understand the empty set if you see that every picture that the brains produces, every experience that the brains is making, is eliminated by the meaning of the definition. In this Zen state this very process of elimination is the only thing left, and then can be seen as the very essence of consciousness itself.
Ayn Rand defined consciousness as an active process of integration and elimination, and distinguished it in this way from passive brain states. (Which many people believe consciousness to be.) Dreaming is a passive brain state, and therefore not consciousness. But nevertheless it is experiencing. So I repeat: when you sleep, and dream, you are experiencing, but not conscious. Consciousness REQUIRES an ‘active state of elimination’.
Sir Arthur Conan Doyle led Sherlock Holmes also express this difference. ‘My dear Watson, you see, but you do not observe’. What transforms an experience into an observation? Some form of symbolic elimination. The house I live in is situated on the second floor. To reach it, I have to climb a stair. I might have done this every day of my life, but if nobody has asked me the question: ‘how many steps does that stair have?’ I would not be able to answer it. The memory of my many experiences of climbing that stairs is just not enough.
It so happens, that after reading this anecdote I have counted it once, and now I know that it consists of 13 steps. And what does this number 13 designate? It tells me that, from my experiences alone, I could not determine how many steps it has. But now that I have counted them, I have ELIMINATED all possibilities of the number of steps it CAN have, and have reduced it to one: ‘a stair of 13 steps, no more, no less.’
The difference between experience and observation is exactly that component we call understanding. All understanding BEGINS with asking questions, which result exactly in only one thing: that what David Hume calls: ‘bunches of specifics’. But in UNDERSTANDINGS, which consist of the ANSWERING of these questions, I am ELIMINATING ON all of these specifics, until a few, hopefully only one remains. We consider a question fully answered, if either one or no answer remains. (Think about trying to solve a 5th degree polynomial with algebraic means. That is impossible, as you very probably know. It solves the problem of trying to find a general formula for solving a 5th degree polynomial.) And we consider that we have some insight, if we can eliminate. What I think of is that astronomers once made a classification of all kinds of galaxies. At one time they recognized that one such class was superfluous, because it consisted of just one type of galaxy, but seen from another angle, making it just appear differently. Such a correction, consisting of an elimination, shows that now there is greater understanding. My point: all understandings are accompanied by eliminations.)
Maybe you might find some clarification of all this in the book of Julian Jaynes: ‘The Origin of Consciousness in the Breakdown of the Bicameral Mind’, wherein he makes a very interesting observation. Julian Jaynes basically claims, that consciousness itself is not something we are born with, but is something that arose out of the emergence of written language. Before written texts existed, so he claims, we had not ONE but TWO speakers in our brains. We had two speech centers, one in the left hemisphere, and one in the right hemisphere. Since the right hemisphere is more directed to the universal, so Julian Jaynes says, this second voice was projected on statues, which were the Gods. These statues were therefore experienced as actually speaking to those people. So before written language the societies consisted of people who were only capable of experiencing, but had no consciousness.
Written texts put such a demand on our brains, that the left-right differentiation we all now have emerged. It eliminated the ‘right hemisphere speaker’. Holy texts replaced the Gods. That is the meaning of the Biblical story, where all statues were destroyed.
So written text resulted in the emergence of consciousness itself. That this is plausible can also be seen from the fact, that many psychologists consider the right hemisphere,which has no speech center as unconscious, and only the left hemisphere, which contains the two speech areas (Broca and Wernicke) as conscious.
Julian Jaynes was the first who identified consciousness as something distinct from experiences. In this book he makes a huge effort to identify consciousness. He fails. But his attempt is interesting, because what he DOES succeed in, is telling what consciousness CANNOT be. So he ends up with a lot of ‘nots’ about consciousness.
He therefore did not make the final step, which allowed me to identify consciousness. It is the ‘notting’ itself, that is, the PROCESS of elimination, which IS consciousness. So if I would describe consciousness in the way Ayn Rand tried to do, I would not call it an active process of differentiation and integration, but simply as an active process of ELIMINATION.
This implies the following: every time we SUCCEED in making an elimination on a whole of possibilities, this FEELS LIKE that we have UNDERSTOOD something. Therefore, understanding is always more than an experience. The ‘more’ consists exactly of ‘elimination’.
This is also what, in my eyes, causes two kinds of mathematics. You have constructive, that is, analytic mathematics, and you have eliminative , that is, synthetic mathematics. An axiomatic system is not universal, because it has left out the details of what it is talking about. No, it is an ELIMINATIVE approach. Axioms are EXPLICIT statements of elimination. They are representations of processes of elimination. And therefore they are attempts to make consciousness itself to become a tool for mathematics.
Oops! I hope this digression was not too long!
My apologies for the delay, Konrad.
This is an interesting line of discussion. The idea that I may be using “experience” in two ways, and equivocating between them, is something that depends on the question of whether the elimination is really something that can be experienced (sensorily), so I will return to the cat-in-the-box example.
It seems to me that you describe a process here, but then want to only refer to the final step of the process; or you describe a movie but then only refer to its final frame:
“What I claim is, that when you understand the meaning of the sentence: ‘no cat in the box’, then what is happening is a picture of a cat in the box, and then an elimination of that cat, so that what is left is a picture of a box with no cat in it, and then understanding that THAT picture is a representation of the sentence: ‘no cat in the box’”
I would say that that MOVIE is the representation of “no cat in the box.” You refer to the “process” of elimination, and I agree with that, but a process is a movie not a static picture. That whole movie stays in our memory.
Also, the static representation (final image of an empty box) is not the same as “no cat in the box,” since the latter is partial information. Sure, we can say the final image has no disagreement with the sentence, but we cannot say the final image conveys the same information. “There is no cat in the box” doesn’t tell us that there is no dog in the box, but going off only the final image of the movie (an empty box) as the complete representation gives us no way to distinguish between “no cat” and “no dog,” or else it is telling us both. Still in neither case is that final image alone telling us the exact same information as, “There is no cat in the box.”
Only the full movie can do that.
If I am correct that this elimination (what I might call a “partial elimination,” not a zen state of complete elimination) is a movie, or at least an image other than simply an empty box (an acceptable image for “no cat in the box” might be a translucent cat in the box), then it seems that this kind of elimination is an experience in the usual sense (a sensory movie). More pointedly, if such partial elimination requires a movie to represent, it cannot also be that merely the image of an empty box suffices.
As to the tangent about consciousness requiring language, it seems that dogs and other animals can perform eliminative processes. Once they look for a bone in a box and find it isn’t in your hand, they will look elsewhere. Though maybe the argument is that dogs do have language, or that this isn’t really an eliminative process?
To contextualize this back to mathematics for anyone else reading this, the initial claim I made was that uttering the term “infinite set” without being able to imagine such a set, is incoherent. Konrad has introduced a point that would, if valid, present a problem for such a simple dismissal of the mainstream notion of infinite set. He presents a possible way out for the mathematician, claiming that it is still possible to understand something even if you cannot imagine it, because understanding consists of both imagining and eliminating. My claim is that such eliminating (that is: partial elimination, not complete zen meditation type of thinking of absolutely nothing) is really all just part of imagining.
It is not enough to say what something isn’t, unless it is plain for the audience to thereby imagine what objects could be included under the definition. For example, it is not sufficient to define an “infinite set” as “a set that is not finite,” because the audience cannot possibly imagine any sets that are not finite. It is all right to define “even number” as “a number that is not odd,” simply because we can easily imagine what numbers this applies to. It is not an acceptable way of defining things in general, but only in certain limited cases.
Reacting to your claim, that elimination is a part of imagination, and also on this: “It is not enough to say what something isn’t, unless it is plain for the audience to thereby imagine what objects could be included under the definition. For example, it is not sufficient to define an “infinite set” as “a set that is not finite,” because the audience cannot possibly imagine any sets that are not finite. It is all right to define “even number” as “a number that is not odd,” simply because we can easily imagine what numbers this applies to. It is not an acceptable way of defining things in general, but only in certain limited cases.”
I think you are very right.
Moreover I think to know where the disagreement comes from.
I have also been thinking about one of your earlier remarks. Namely, that the very essence of abstraction consists of only elimination of certain parts, so that what is left is applicable to more. (And therefore, you HAVE to conclude, abstraction is impossible without some form of imagination still to be left over. I see that it is THIS where our disagreement comes from.)
To give a very simple example of what I think you meant in set theoretical terms, the set of all red apples is smaller than the set of all apples. So the set of all apples is ‘more abstract’, because there are more things that are just apples than there are things that are both apples and red.
So the less attributes you use as a collection of defining properties in mathematics, the more it will be applicable.
Now this is irrefutable, because it is part of logic. p & q is less specific than p. But you use it to DEFINE abstraction. You think that this is ALL what is abstraction about.
And it is exactly here where you are mistaken.
Some years ago I have studied a very interesting book about geometry. It is the book: Geometry of David A. Brannan, Matthew F. Esplen and Jeremy J. Gray. (Cambridge University Press, ISBN 0-521-59787-0.) In it they made clear (to me, at least) that there is another part of abstraction that definitely has to belong to it. What they did in this book was simply to ask what a triangle is,and to define this as precise as possible, given the fact that the set of all triangles is infinite. How can our mind grasp this? This problem is much like what we are discussing here, although ‘the infinite’ was only implicitly present in their approach. (They took infinite sets for granted.)
The answer they gave was that you have to give ONE CONCRETE triangle, and then you have to attach the GROUP of all AFFINE TRANSFORMATIONS to this one triangle.
The reason why this works is because it is possible to transform any triangle into any other through an affine transformation.. And this allows us to specify ANY triangle by giving just one triangle and this whole set of all affine transformations. Or, rather, this GROUP of affine transformations.
Interestingly, you can also characterize the ellipse, the parabola and the hyperbola in this way. That is, take just one particular ellipse and another arbitrary ellipse, and then there will be an affine transformation that transforms your chosen ellipse into that other, whichever it is. So the concept of ‘ellipse’ can be captured in this same way.
So, through this group of affine transformations you know what to include if you consider any infinite set of ellipses, and therefore the concept of ‘ellipse’.
But how about EXCLUDING everything that is NOT an ellipse? How do you exclude a parabola or a hyperbola? Those are, after all, also quadratic forms.
Well…
The book showed, that it is impossible to find an affine transformation that transforms an ellipse into a parabola, a parabola into a hyperbola etc. And that deals EXPLICITLY with the EXCLUSION part of the abstraction.
BTW, in the text above you only mention inclusion, but not exclusion. This is my critique on your definition of abstraction. If you define abstraction only as elimination, then you arrive at a criterion with which you can include, but not at something you can use to exclude. That is why I object to your definition of abstraction, and find the approach of Brannan et. al. so much better.
To continue with exclusion, and another interesting case, you cannot transform a triangle into two parallel lines through an affine transformation. So not ANY set of three lines forms a triangle.
(BTW, Any affine transformation of the plane into itself can be described by an equation of the form t(x) = Ax + b, whereby A is a 2 x 2 matrix, and b is a vector..)
So, with this approach you can give a definition of the concept of triangle, ellipse, parabola and hyperbola. In fact, in this way you ‘capture’ the propertyof ‘ellipse-ness’ by considering it as an INVARIANT of the group of affine transformations. The same applies to triangles, parabolas and hyperbolas.
In other words, the abstract nature of an ellipse is not captured by elimination alone, but also by something extra. The fact that we can transform any ellipse into any other allows us to come up with one concrete ‘model ellipse’, so to speak, and to ‘use affine transformations to recognize’ the ‘ellipse-ness’ of any other ellipse, by showing the existence of an affine transformation that transforms my one ellipse into any other that is presented by me.
Interestingly, with this approach you can go further. Suppose, for example, you want to have a clear definition of a quadratic form in general. What you then need is a way of transforming an ellipse into a a parabola or a hyperbola etc. You can do this, by introducing homogeneous coordinates, and then considering the group of invertible 3 by 3 matrices. What you then form is the group of PROJECTIVE transformations. With this group you are able to transform any quadratic form into any other. In other words, this group DOES NOT DISTINGUISH BETWEEN different quadratic forms. Moreover, with this group you can transform a triangle into a figure consisting of two parallel lines intersected by a third.
Seen in this way, abstractions do not just consist of eliminations and ‘seeing that the less properties I consider, the more things it will include’. No, there is another aspect of abstraction. Namely by connecting them to groups of transformations, that is, to SYMMETRIES.
This approach makes the very process of abstraction precise. Apparently, what we do when we are abstracting (within one context) is the following.
What we need are at least THREE examples. Why three? Take the simplest example above. I want to define a triangle. What I need is two triangles that I can transform into each other, and a third object that is NOT a triangle, to know what to exclude. This third object can be two parallel lines intersected by a third. So it is an object that consists of three lines and just two intersection points, showing that not any three lines will form a triangle.
I think this is the way abstraction occurs in general, at least when we restrict ourselves to one context. To FORM abstractions you need concrete examples. In that I agree with you. But also,, you need at least three examples: two that correspond to your abstraction and one counterexample that tells you what you must exclude. Only then you have the minimum required to form an abstraction.
My point is, that an abstraction is not just elimination, and recognizing concrete instances of abstraction is not just imagination. Abstraction consists of a process of inclusion and exclusion, leading to not just a set, but also to a group of transformations that allows us to go to any concrete example of an abstraction to any other, but also definitely NOT to the counterexamples. Therefore, to construct an abstraction you need at least two concrete examples and one counterexample.
Now my point is this. AFTER you have constructed the abstraction, or, AFTER our mind has formed the abstraction, you can ELIMINATE the three concrete examples. What then is left is just the group of transformations itself.
This group of transformations therefore is the realization of the ABILITY to both include and exclude. It is what, in our mind, gives ‘sense that we understand’, that is, ‘the feeling of understanding.
But if that is so, then understanding and imagination are different things. Indeed, THROUGH imagination we ARRIVE AT understandings, but imaginations and understandings are not the same thing.
Immanuel Kant began his ‘Critique of Pure Reason’ with: ‘Although all understanding begins with the senses, it does not arise out of the senses’. In the same manner I claim that all abstraction begins with imaginations, but it does not arise out of imaginations. Abstractions are the result of an active mind, that tries to create a group of transformations out of concrete examples, both examples and counterexamples, which, when it succeeds, results in something in our mind that gives us the ability to both include and exclude. But this ability, once acquired, can then dismiss the concrete examples out of which it arose.
I want to point out one thing more about this. If you read some of the books of 19th century and 20th century mathematics, you can find people who are proud of their ability to write texts in which they show their understandings without any pictures at all. Lagrange, for example, was proud of the fact, that he could write a complete book of mechanics, without even one picture.
I think they went way too far with that. AFTER you have formed your abstractions, you can dismiss the concrete imaginations. But to FORM them you NEED concrete imaginations. Therefore, to communicate abstractions, you have to give concrete examples.
So Ishinden. I hope you ‘take that what I have presented here in’. I think the book Geometry gives a foundation of what I am telling about. And I think that, if you go through that book, you have a very clear understanding of my point. I see your point, but I think you do not see mine.
Little mistake. I wrote: ‘p & q is less specific than p.” Must be: p&q is more specific than p. Or: p is more general than p & q.
Hi Norman,
I think it will be more interesting if we ask a modern mathematician to prove that pi+e is equal to e+pi or pi*e is equal to e*pi since he or she believes that real numbers satisfy the laws of arithmetic and see what he or she says. Perhaps, the answer will be philosophical, not mathematical.
It would follow fairly directly from their series representations, and since they allow themselves an infinite number of operations they’ll finish it in no time at all.
But even if you allow not an infinite number of operations, you are able to say the following. Approximation of pi up to n
pi = 4 (1 -1/3 + 1/5- 1/7 + … (-1)^(n+1)(1/n))
approximation of e up to n
e = 1 + 1/2! + … + 1/n!
Therefore approximation of pi + e up to n becomes
[1 + 4] + [1/2! -4/3] + … [1/n! +(-1)^(n+1)(4/n)]
which is also an approximation of e + pi up to n, because of commutativity of addition for fractions.
Now, if we assume that only a finite number of terms exist, BUT we can extend the number of terms whenever we want. (That is what the meaning is of the fact, that the formulas above contain an expansion up to n, and not an expansion up to, say, 1000.) That is, the appearance of the variable n means that the number of POTENTIAL TERMS are NOT FINITE, then there is NO REASON WHY pi + e is different from e + pi. Therefore, logically speaking, from the fact that both series contain an indeterminate last term, expressed by the symbol n, we can conclude that pi + e IS NOT UNEQUAL TO e + pi.
Saying that pi + e is not unequal to e + pi is not the same as saying that pi + e is EQUAL to e + pi, in the same manner as saying that something is NOT FINITE is not equal to saying that it is INFINITE. Nevertheless, if we assume the validity of the logical law of excluded middle, this is tantamount to a proof of the equality of pi + e and e + pi.
Norman, I have been thinking more about your thesis of infinity and rational numbers. . You equate the set of real numbers with the set of decimal representations of real numbers, and from the fact that numbers like Sqrt[2] require an infinite expansion to be represented uniquely you derive your objections.
But if that is so, then even fractions do not escape this objection. Take, for example, the fraction 1/3. In decimal representation this requires 0.3333…, which is also an infinite expansion. Or, 1/7 = 0.142857142857….
Only fractions of which the denominator is divisible by 2 or 5 have finite expansions. But does that mean that only fractions that have denominators that are divisible by 2 or 5 are the only fractions that are allowed? And if not, are all infinite expansions that have a repeatable pattern allowed? (Which, indeed, include, or, better, is equal to the set of all fractions.)
But then there is no reason to exclude Sqrt[2] and the likes, because although it has no repeating pattern in a decimal expansion representation, it DOES HAVE a repeating pattern in a continuous fraction representation:
Sqrt[2] = 1 + 1/(2 + 1/(2 + … )))
In fact, any square root, or any number of the form (a + sqrt[b])/c can be expressed as a repeating continuous fraction representation. And therefore, if we also allow the continuous fraction representation, there is no reason why square roots should be excluded. In other words, square roots are as exact as fractions are.
But even more is possible. I have followed a course in numerical analysis. In it I learned an interesting principle. Within numerical analysis, a problem is considered to be solved whenever you can find an algorithm that is capable of ‘zooming in’ on the solution to a problem at an arbitrary precision. Now, we DO HAVE algorithms to calculate e, pi, and many other real numbers up to arbitrary precision. This means, logically speaking, that such numbers are, under the criterion of numerical analysis, well – defined. And therefore there is no reason why even these should be excluded from our set of numbers we use as a basis for mathematics. In fact, under this criterion, and from the fact that we can give a rational expansion of e and pi, up to an arbitrary n, it follows that these numbers are also as exact as the rational numbers. The logical principle is: whenever it is possible to distinguish a particular number from any other number different from it, (both being clearly defined under the criterion of numerical analysis), these numbers are well-defined.
What this DOES RESULT IN, is an ALGORITHMIC definition of numbers. All numbers that can be defined algorithmically ‘exist’ in the sense you use the term.
Dear Norman.
I am following your course on differential geometry. What a beautiful approach to the differential calculus! I have been experimenting with it, and with Geogebra and with Mathematica. What a beautiful way to avoid limit processes, through replacing them with algebra! A huge advantage of this method is that you do not have to form approximations from differentials by a separate thought process. Although it is algebraically quite involved, and therefore in practice it can only be done by using mathematics programs, it is conceptually very neat!
By the way, did you know, that they have added 3 dimensional graphic capabilities to Geogebra 5s?
In any case, if you insist on avoiding infinities, then, as I have said in my last e-mail to you, you also have to avoid the set of rational numbers, because as a SET it is infinite too.
The only way to avoid the infinite in a consistent way, is to allow only for finite sets. And if you do, then there is no problem with introducing square roots of 2, 3, or any integer, including the negative one’s.
This is because the only problem you have, is by allowing the infinite as an actuality.
I agree with you, that our minds are not able to imagine the infinite as a finished structure. Moreover, if you look very closely to the very concept of ‘infinite’ as it is used in mathematics, then in each and every case what they REALLY do, is ASSUMING that something is finite, and then derive a logical CONTRADICTION from this. This means, that what they are proving is not that something is ‘actually infinite’, but that something is ‘not finite’.
This is important, because a logical contradiction not only proves that something does not exist, but it can also be a proof of the opposite: namely that besides that one thing, there can be many things that exist that have the same property. Therefore the logical denial of something is not equivalent to the logical establishing of ONE other thing. If I prove, for example, that some animal is not an elephant, then I have not proved that therefore it IS an infinite elephant. No, I have only proved that it can be many other different kinds of animals, beside the elephant.
In the same manner, if I have proved that ANY particular set of integers is NOT finite, then the only thing I have proved is that there can be at least one set that is larger than that particular set.
If I have not specified the particular set of integers, and only have assumed that it is SOME particular set of integers, then the proof that the set is infinite can be applied to ANY set of integers. And that means that the only thing that I have proved is that there can be A set that is larger than whatever particular set I focus on. So what I have proved is basically this. Suppose I take a finite set of integers, say In. and the largest integer in that set is N. Then I can construct a set from In that contains at least one integer that is larger than N, for example, by taking the union of In with the set containing N + 1. In other words, I construct In’ = In U {N + 1}. This then, leads to a set that is as finite as the original set, only it has one more member.
The way I can word this is by stating that a proof that leads to a contradiction when I assume that it is finite, leads to the conclusion that it is POTENTIALLY EXTENDIBLE to a larger set.
And from this set I can construct a Qn, a set of fractions that is also finite, by defining
Qn={ x = a/b | a E In & B E In\ {0} }.
Moreover, I can construct a finite set that contains square roots, by constructing, from In, another set containing the square roots of In, let us call it Isqrt{n], and making a union between In and Isqrt[n], that is Iq[n] = In U Isqrt[n].
This set is finite also, because the number of elements it contains is N^2. From this set it is possible to construct
Qq[n] = { x = a/b | a E qIn & B E Iqn\ {0} }.
Moreover, the fact that the decimal expansion of, say, Sqrt[2] is not finite, is no problem in this set, because you can place all elements of Qq[n] on one number line, or, if you allow for negative integers, in a number plane where every one has a definite location.
To show what I mean, suppose I choose for my set of integers the set I1000+, the set of positive integers from 0 to 1000, and I construct a Qn out of this. Then the total number of elements in I1000+ is 1001. I can then form a new set, consisting of all square roots of the numbers 1 to 1000, and call it Isqrt[1000], and add this to In. Then I can construct a Qsqrt[1000] in the above manner.
If I construct Q1000+, then the total number of elements this set contains is equal to 1000 001 elements, to allow for all positive fractions. (Each decimal of the form 2.12 can be placed in that set, because 2.12 = 212/100 = 53/25. Moreover, I need at least 1000 integers to be able to represent any fraction in a decimal representation containing two decimal places in the set Q1000+. If I want to have Sqrt[2], then I just add it between 141/100 and 142/100, so that I can see how large it is, compared to the other elements in Q, but do nothing with it. I just recognized it as an element that has the property that when it is squared with itself, it produces 2.
In other words, it is quite feasible to have finite sets, and to include square roots. In fact, with this method I can have finite sets that include roots of any power. Also, the addition of Pi, e, etc. is also not a problem. I place Pi between 314/100 and 315/100, and use as its defining property that it is equal in magnitude to the area of the unit circle. And I place e between 2.71 and 2.72 and use as its defining property that it is the base of a function whose change is equal to the function itself.
The advantage of this approach is that all sets are finite. And any algebraic or transcendental number can be added to the construction of such sets. Moreover, any number that is not an integer or a pure fraction is defined by its most characteristic property. This has the advantage that whenever they appear in mathematical formulas I can already see some connection with mathematical structures. It has the further advantage that I make the context of any mathematical structure precise, so that the theories of mathematics are stated with far greater precision.
So although I fully agree with your idea that the infinite does not exist, I do not agree that this leads to the abandonment of sqare roots, or any other number that is already used within mathematics. Moreover, Your approach is both too restrictive and too wide. Too restrictive, because it leads to the abandonment of square roots and the likes. And it is too wide, because even if you restrict yourself fully to rational numbers only, you are still dealing with the infinite sets of integers and that of rational numbers.
Thanks for a very interesting comment. There is a lot to ponder here, and many worthwhile directions for thinking. I do believe that we should consider doing arithmetic with the integers replaced by, or considered to be, some large finite set. One other approach to setting this up which you did not mention would be to work with some largish prime p, and then work mod p. That would avoid approximations or truncations once we got to the size of p, but would introduce wrap around.
Your general approach is that of someone familiar with computing. I believe this is the direction that pure mathematics needs to move to the next level in its search for foundations.
Another approach to square roots, of course, is to consider them in terms of extension fields. In this approach, sqrt(2) is not something between 1.4 and 1.5, but an entirely new symbol with prescribed arithmetical properties.
You wrote: ” In this approach, sqrt(2) is NOT something between 1.4 and 1.5, but an entirely new symbol with prescribed arithmetical properties.”
Why exclude the possibility that it is something between 1.4 and 1.5? Why can’t it be both? Why not see Sqrt[2] as a symbol with prescribed arithmetical properties, and if you want to have a sense of the magnitude it represents, see it as something between 1.4 and 1.5?
That was my point in my e-mail to you, where I talked about representations within quantum mechanics. In QM you have the momentum representation and the position representation. In QM this means, that you can have complete knowledge of the position of a particle, at the cost of knowing nothing at all about its momentum, or the converse. If, however, you give up complete knowledge of the position, and accept to know it up to a certain interval, you can have knowledge of an interval of the magnitude of the momentum.
More interestingly is the way you can calculate within QM. Usually they set up the differential equations within the position representation. This is because if you use the momentum representation to set up the differential equations describing the wave function, they become almost impossible to solve. (Technical detail: this is because all measurable magnitudes within classical physics, when converted into QM equations become operators. In particular, the potential becomes an operator. In the momentum representation you must replace p, the momentum by -ih dx/dt, and you can replace the position operator by just x, the position itself. This does nothing to the potential, since this contains only the position. (When the potential is time-independent.) But if you choose the momentum representation, you can replace the momentum operator just by p, the value it has. But then you MUST replace the position operator by -ih d/dp. This transforms the potential as a function in most cases into a horrendously complicated potential operator, which leads to far more difficult differential equations to solve.)
This brought me on the following idea. Numbers in general are used to calculate certain properties. Magnitude is just one of the properties a number can represent. To do that they must be able to stand for those properties. And the properties they can have are dependent on the underlying set they are part of. The type defines the properties, and then you can use that to construct a set that is large enough to capture all magnitudes that are relevant, or that can occur in some experiments.
So, why not go from properties to sets, and then see from the properties what you can calculate with those sets? In that sense, the underlying set looks much like the different representations of the magnitudes used in quantum mechanics.
This means, that you can see sqrt[2] as a symbol with certain prescribed arithmetical properties. These properties then define a set where sqrt[2] is one of its elements, and which can contain many other elements of the same kind. The type of the set then defines what the set represents. The concept of Magnitude then leads to the set of a limited number of decimals, which is most suited for those things usually used for real numbers.
Seen in this way, the fact that you need an infinite decimal expansion to represent sqrt(2) as a real number can be seen as a problem of representation, and not as a token, that you can only define sqrt(2) through accepting the infinite in mathematics. In particular, it is caused by the fact, that you want sqrt[2] to stand for a certain amount.
I have followed your lecture on projective geometry. There something similar exists. Within Eucildean geometry they say, that two parallel lines do not meet, no matter how far you extend them. Later they ‘understood’ that as meaning that the two lines ‘meet at infinity’. (Again, ‘not finite’ is interpreted as ‘infinity’.) But you can construct the projective plane, and then you can SEE where they meet. In essence you have changed the representation. And in THAT representation the two lines DO VISIBLY MEET. That is, you do not need infinity to see that any two lines meet in a point. What you give up, of course, is the metrical properties of Euclidean geometry. So you see, that the ‘meeting at infinity’ is the result of insisting on the Euclidean metric. But by just eliminating the metric, as is done in Projective Geometry, or even by just introducing a different metric, like is done in spherical or hyperbolic geometry, you can eliminate the infinities, and still transform between different representations. In that sense, the infinity is seen as a manifestation of representations, and not as a token, that there is something that is ‘really’ infinite.
Something else might also be interesting. In propositional logic you have so-called ‘axiom schemes’ . The problem there is, that there is a connection between tautologies and logically valid arguments. Any logically valid argument can be transformed into a tautology. And any particular tautology stands for a whole set of logically valid arguments.
This makes the concept of ‘tautology’ very interesting. Moreover, logicians were aware of the power of axiomatization. Therefore, they attempted to axiomatize the concept of tautology. And when they did, they discovered, that you needed an infinite number of axioms to build a system from which you can derive every tautology.
Faced with this problem, they came up with the concept of ‘axiom scheme’. That is, a formula that tells when something is an axiom within propositional logic. In some systems you need 3 axiom schemes, which produce an unlimited number of axioms.
But by a simple ‘change of representation’, through going to second order logic, you could just eliminate this infinity.
So maybe you should consider infinities in mathematics in the same way as infinities in physics. Whenever there is an infinity in physics, it is not seen as an actual infinity, but as a token that the theory itself has some problem, and therefore you must search for another theory that is able to describe the same aspect of reality, but with the infinity eliminated.
In the same manner. If you are confronted with an infinity in mathematics, then this should be seen as a sign that the mathematical theory has a shortcoming, and that you should be looking out for another representation of the same field of enquiry, in which the infinity is eliminated.
And, as a last remark. The idea of different representations of numbers is not entirely new. I got it from Paul Dirac, who talked about ‘p – numbers’ and ‘c – numbers’ within QM. He only applied it to physics, though.
I did not realize, that my training in computer programming could shed light on mathematics itself. Thanks for pointing this out.
In any case, thinking about the concept of type, as used in Delphi and CBuilder, I saw that there was another concept that might be helpful.
If you make a computer program, you begin with types. Then you declare variables. Then you initialize them, which causes the computer to assign memory space to these variables. And then you put data in those variables, which the program operates on through the statements.
One of the greatest problems of programming, maybe THE problem, is that you have to code in such a way, that YOU, the programmer, is able to read your own code after several months, and see exactly what the how and why of your statements are. Mistakes in programming are very easily made.
Therefore, computer language designers have introduced another kind of magnitude, called the TEMPLATE.
When I began with programming, I had not a clue where these templates were for. Later I realized, that they were only there as constructs to avoid mistakes.
What is a template in programming? A template is some definition of a class (type) that CANNOT be instantiated as a variable. They are something ‘above’ types. When a programmer wants to define a type, and wants to avoid mistakes, he first creates a template. And then he defines a type corresponding to the template. The function of the template is therefore to prevent mistakes when the programmer is defining his types.
The way to understand this, is by looking at the concept of context. In order to define something, you have to have a context. A context is a ‘possible world’. WITHIN that possible world you create definitions. A definition consists of some word, and a definition of what the word means. And ‘meaning’ consists of an ability within some context to make a distinction between something does, and does not satisfy that definition.
To give a simple example. Take ‘even number’. To define ‘even number’you begin with some set of integers, say In. And then you look at each integer in that set, to determine whether it is divisible by 2. And to define THAT exactly, what you do is you look at every element in In, divide it by 2, and look whether the result of this division is still in In. If it is, you say that the original integer is even, and if it is not, you say it is not even. In this way you see that there is a distinction within In between two kinds of numbers: even and not even. And BECAUSE the concept of ‘even number’ creates this distinction within In, you say that the concept of ‘even number’ makes sense in the set In.
However, if you EXTEND In to a set consisting of rational numbers, in the following way:
Qn ={ x | x = a/b, a E In & b E In – {0} }
and you try to apply the above definition of even number to this set, you will see that the result of EVERY division by 2 of an element of Qn results in another element of Qn. Therefore, within Qn the definition of even number does not cause a distinction within Qn between two types of numbers, and therefore this definition ‘does not make sense’ in this set.
This shows, that concepts and definitions are dependent on the sets within which these definitions are applied. I consider such a set to be a CONTEXT. Therefore I word this by saying, that the function of a concept is the ability to make distinctions within contexts.
This is also the function of a type. A type is, in essence, a definition. It presupposes the existence of the context. But that leaves us with another problem: how to construct the contexts themselves. And that is where the concept of TEMPLATE comes in. A template is something that is used to see, whether the context itself is well-defined. It can be used to see whether you can create a context that allows for definitions in such a way, that they satisfy the criterion of ‘the ability to use them to make distinctions’, so that logical thinking as such does function in such a context.
In this sense, the rules of propositional logic and predicate logic are the templates of mathematics itself. They tell us whether some theory of mathematics is sound.
Seen in this light, your observation that there is something seriously wrong with the whole concept of ‘real number’ can be fully clarified. For what you are saying is that if the definition of Sqrt[2] within the context of real numbers REQUIRES decimal expansions that are completely irregular AND infinitely long, AND it is the case that we are unable to imagine anything infinite, then this means that defining Sqrt[2] by its infinitely long decimal expansion does NOT satisfy the criterion of ‘the ability to distinguish, within a context, something belonging to a definition, from something not belonging to this definition’.
My response to this claim is that you are right with your claim, that within the context of real numbers this is an unsatisfactory definition. However, and that is what I claim, it IS possible to construct the context, the underlying set of numbers in such a way, that such a definition IS possible.
So the discussion is one about templates. The concept of ‘template’ is another example of what I have called ‘potentialities’. A ‘potentiality’ is some description of something that does not exist, but can be MADE to exist through construction. So ‘Integer’ is not a type, but is a template in this language. We define contexts through templates. And within those contexts we define definitions that allow us to make distinctions WITHIN those contexts. So with a template we can see whether something belongs to a certain context. And with a type we can see whether we can make distinctions WITHIN such a context.
Small error. I wrote ‘the momentum by -ih dx/dt,. Must be: -ih d/dx.
Maybe a nice afterthought. If infinities do not exist, the exact converse, the so-called empty set also does not exist. But is the fact, that some argument leads to the empty set not equivalent to discovering, that ‘there is something that does not exist’?
But that is a contradiction. How can there be ‘something that does not exist’?
Therefore, the way to understand this is, if you discover, through some logical argument, that some set is empty, then this is equivalent to the statement that that set does not exist. Therefore, the symbol of the empty set stands for ‘non-existence’ as such, and NOT for ‘something that does not exist’.
This also means, that von Neumann’s method of constructing the integers:
0, {0}, {0, {0}}, {0, {0, {0}}}, … } is nonsense.
You often say that we should look at the ancient Greeks, and what they said. In ancient Greece they made a distinction between magnitude and number. A magnitude is something continuous, while a number is something discrete. Geometry used to be about the continuous, while number theory about the discrete. Descartes tried to bring those two together, by connecting numbers and geometry with algebra. He disregarded the fundamental difference between the discrete and the continuous.
I think that the origin lies in talking about points and lines. A point is something discrete. A line is something continuous. By talking about lines as sets of points you make an error. This is because a SET is always something DISCRETE,while a line is something continuous. Therefore you cannot say that a line is a set of points.
What I suggest is to replace the idea of a point with the idea of a position. It might look like only a minor change, but it restores the concept of a line as a continuum. What you then must say, is that a line is a continuum of positions.
As I said, the difference might look small, but it makes more sense in some problems. Take, for example, the cutting of a disk into two equal halves, Within set theory this is not possible, because if you cut the disk D into the two halves H1 and H2, the middle point either goes to H1 or to H2, so that the symmetry is lost. But if the disk consists of a continuum of positions, then the center position is present in both halves.
You must change the language somewhat, which becomes even simpler. Two non-parallel lines (in Euclidean geometry) intersect each other in one point becomes: two lines meet each other AT one position. Two parallel lines have no position in common etc. The reason why you need such a language is because if you insist on a decimal representation of all numbers, then it is very unlikely that two arbitrary non-parallel lines will intersect at one point. This is because this can only be true if the point of intersection is represented by some pair of rational numbers, or some pair of decimal representations. And then you have the problem that decimal representations are not unique.
This is because 1/3 = 0.333…, 2/3 = 0.666…, and 3/3 = 0.999…. = 1? Really? Is there a difference between 0.999…. and 1? I think there is. And if that is so, then you can never say that two non-parallel lines intersect at a point. But if you assume the continuous as something separate, which is independent on any representation, then you CAN say that two non-parallel lines MEET at one POSITION. Moreover, you can only make this claim if you use synthetic geometry. Also, if you want to apply algebra to geometry, then you must not interpret the letters in the equations as numbers, be it rational, real or otherwise. This is because numbers are ALWAYS something discrete, while geometry does not consist of discrete numbers, but consists of continuous magnitudes.
So the equation of a circle x^2 + y^2 = 1 describes positions, not points. And as such it can be applied to numbers. Therefore I suggest the following name giving. As long as the equations of algebra are applied to geometry, they do not correspond to numbers in any way. They describe POSITIONS.
However, if you substitute numbers in the symbols of algebra that represent the magnitudes of geometry, then you are transforming continuous magnitudes into discrete numbers, which always imply an approximation. Even in the case of integers and fractions. I suggest that if you do, the SETS of NUMBERS you then get are POINTS.
So I do not disagree with Descartes, that you can transform geometry into algebra. But I DO disagree with interpreting the algebra equations as sets of numbers. So I think it is impossible to transform geometry into number theory. Algebra and geometry are at the exact same level of abstraction. But numbers are one step more concrete.
After reading Descartes original work it is much clearer what he had in mind. I would even argue that he had more mathematical common sense then many mathematicians that work today.
I agree with the idea that the marriage between algebra and geometry is the proper way to treat problems of analysis. That is because continuum does exist in geometry. So if we have a segment of length x0 and f(x) is any standard function that we use then there is a segment of length f(x0). This of course may not be expressible in numbers.
However in his work La Geometrie Descartes DID NOT use a coordinate system and DID NOT attach numbers to points but in fact used letters all through the work. If anyone can direct me to the actual work in which coordinate system was actually used in the modern sense by Descartes I would really appreciate it.
Yes, Aleksandar. I quite agree with you. But you can go even one step further. (Meaning, I begin to doubt Norman Wildberger’s attack on the infinite.)
Norman Wildberger has given himself, in one of his last video’s, an interesting first step showing that there is a connection between the actual infinite and the continuous. It is the famous example of Zeno, of Archiilles who is unable to overtake the tortoise.
Seen in that light, let us make it more concrete, so that you can see why I begin to doubt him very much.
Suppose we have Archilles, and the Tortoise. Let us assume, that the race track is 1 kilometer long. Let us also assume, that it takes, say, Archilles 1 minute to cover the whole track, while it would take the tortoise 2 minutes.
Suppose the tortoise gets a headstart of 1/2 kilometer.
This means, that when Archilles has covered 1/2 a kilometer, the tortoise has covered 1/4 kilometer. And when Archilles then runs for 1/4 kilometer, the tortoise has run 1/8 kilometer.
So the distance between Archilles and the tortoise begins with 1/2, then becomes 1/4, then 1/8 etc. And IT HAS BECOME ZERO when Archilles has run 1 kilometer, and the tortoise has run 1/2 kilometer.
THIS WILL HAPPEN!
The total distance Archilles will need to cover the whole distance of 1 kilometer is the following sum: 1/2 + 1/4 + 1/8 + 1/16 + … + 1/2^n + …
And the total distance the tortoise will need to cover the whole distance of 1/2 kilometer is
1/4 + 1/8 + 1/16 + … + 1/2^(n-1) + …
Now, according to Zeno, Archilles will never overtake the tortoise, BECAUSE the sum of an infinite number of distances, no matter how small, will always be infinite.
We know since at least Leibniz time, that the sum
1/2 + 1/4 + 1/8 + … + 1/2^n = 1, and 1/4 + 1/8 + 1/16 + .. + 1/2^(n-1) + … = 1/2.
So we know that Zeno was wrong. And we know he was wrong, because we can all see that Archilles will definitely overtake the tortoise in this situation.
From this assumption we can begin to think, and then arrive at the conclusion that Zeno has made an implicit assumption which makes that his argument appeared to be plausible, but which was wrong. (Because it was refuted by reality) This implicit assumption is that if we add an infinite number of numbers, no matter how small, and no matter how fast they become smaller, this sum has to be infinite. But we can simply SEE that Archilles WILL OVERTAKE THE TORTOISE. And therefore we can CONCLUDE that this assumption is wrong.
But THAT means, that we KNOW, FROM OBSERVATION that infinite sums that lead to a DEFINITE outcome DO ACTUALLY AND IN FACT EXIST. We can simply OBSERVE that this is the case, by seeing that IN REALITY Archilles WILL OVERTAKE the tortoise.
But that means, that the assumption that the argument that infinite sums do not exist, IS SIMPLY REFUTED BY OBSERVATION.
And therefore, Norman Wildberger, with his attack on real numbers, is mistaken!
But not all is lost. He made me see, that whenever we are confronted with an infiniyy leading to a finite result, there migth be an underlying continuum that integrates them. So the infinite is the result of a limited representation. That is, as soon as you try to represent the continuum with something that is fundamentally discrete, you will end up with some form of infinity. This infinity may not be understood in terms of something discrete, but it CAN be understood in terms of something that is continuous.
Seen in this light, I see Norman Wildberger as a present day Zeno. Or, better, as a present day Brouwer. Norman Wildberger has started up the discussion which existed between Brouwer and Hilbert, which ended then inconclusive, and therefore was not solved.
And the reason why it was not solved then, is because of something Henri Poincaré said. He asserted: ‘existence is consistence’. And that is a mistake. If you can make a system that is consistent, then it does not necessarily have to exist. That is why non-Euclidean geometry was seen as something not real, until Einstein came along, and showed that his general theory of relativity required non-Euclidean geometry, even when you only consider space. He showed that you could find relations in existence that can only be adequately understood in terms of non-Euclidean geometry.
So before Einstein non-Euclidean geometry was seen as a logical possibility, but not as an existing possibility, although Gauss tried to devise experiments wherein he investigated whether a form of non-Euclidean geometry could be applied to existence. After Einstein, it was no longer just a logical possibility, but it became an actual fact. The world ‘really is’ like that.
The discussion reminds me a little bit about another famous problem of mathematics: to determine whether something is computable or not. It was seen as a purely mathematical problem, until David Deutsch said, that we had it all wrong. The question about computability is not just a mathematical problem, but also belongs to physics. So computability depends on the actual existence of certain entities capable of computation. Such entities had ‘the physical property of the capacity to compute’.
In the same manner, plane geometry was, since the culture of ancient Greece seen as a purely mental discipline. Then Newton made the remark, that plane euclidean Geometry should be considered not just as mathematics, but as a theory of ‘natural philosophy’, which was his name for physics. (The term ‘physics’ did not exist yet then. What he meant was that plane geometry is not just mathematics, not just a matter of the intellect, but is ‘imposed upon us’ by reality itself. So it is not only a theory of mathematics, but it is a form of physics. He even added, that geometry is the part of mechanics, that deals with measurement. Others extended this remark by stating, that mechanics consists of geometry, to which time is added.
Seen in this ligth, we might THINK on logical grounds, that the actual infinite does not exist. But our OBSERVATION shows us, in the example of Archilles and the tortoise, that the actual infinite DOES EXIST. Moreover, matters pertaining to the infinite do not just belong to mathematics, but they belong to physics as well. And THAT means, that we can try to develop a symbolic framework whereby we deal with the infinite.
In fact, that is exactly what was done in the 19th century by Cauchy, Riemann and the like. And exactly what Cantor also did, but more abstractly.
David Deutsch remark is wider in scope. His unusual approach is to consider human beings themselves as an example of physical objects capable of computation’, that is, of symbol manipulation. Seen in this light, the correct question is not whether the actual infinite exists (it does, because Zeno’s and the tortoise example shows this) but to ask ourselves whether we can capture the infinite in sequences of finite symbols. So we should ask ourselves whether, whenever we encounter an infinity, whether our symbolic approach to reality is limited, and in what sense, exactly.
The equation of a circle x^2 + y^2 = 1 captures a continuum. The infinite arises when we try to apply some system of symbols on it, that is only capable of expressing ALL particular solution to this equation with a finite number of symbols. Such a system always fails. And therefore we express the totality of solutions of this equation as ‘not finite’, from any such system, which is then imagined to be ‘actually infinite’. The ‘actuality’ is correct. But the ‘infinity’ part arises out of our representation. Most number systems, because they can only expresss one element of such a set at a time, express the solutions with a finite number of symbols, is not capable of doing this. And the reason is not that these elements those infinities try to ‘get to’ do not exist, but that any symbolic system that is only capable of ‘expressing just one point at a time’, is not capable of exhaustively expressing any solution of such an equation..
In other words, questions pertaining to the infinite are REPRESENTATION PROBLEMS. They show up as inadequacies of our symbolic approaches to reality. Apparenty we are always faced with the following choice. Either we try to devise a system of symbols that is capable of expressing as many solutions of such equations as possible, with the understanding that there will always be solutions that cannot be expressed in this way. Or we insist on a symbolic system we claim to be complete, and therefore capable of expressing each and every solution of such equations, and DENY the existence of solutions that cannot be expressed with our symbolic system.
Norman Wildberger, with his insistence on the rational numbers, chooses this second option.
I find this too restrictive, because if you take this position, you are going to attack anything that points at the existence of those solutions, that cannot be expressed by your system of choice.
And that is exactly what Norman Wildberger is doing. Not that he does not have a point, but it is a different one than he imagines. What Norman Wildberger’s attacks mean to me, is that he has made me aware of the fact, that our symbolic systems are always limited in expressing any continuum. As such his contribution is huge.
In the book: ‘Methods of Theoretical Physics’ of Morse and Feshbach you can also find an interesting remark about singularities, the name physicists give to infinities. They assert, that whenever you encounter a singularity in your theory about the world, in the theory of physics, this shows that the theory is incomplete, and therefore asks from the investigators, to try to come up with a better theory in which the singularities disappear.
In the same manner, whenever you encounter an infinity in mathematics, then this does not prove the non-existence of that what the mathematics tries to express, but it shows that there has to be a better mathematics, in which the infinity disappears.
And if you look at it this way, then it is often doable. Norman Wildberger’s very own Rational Trigonometry is an attempt in this direction. He saw, that if you express the magnitude of the hypothenuse of a right angular triangle with two sides having two unit lengths with just taking lengths as the basic magnitude, then the length of the hypothenuse will be sqrt(2), which has an infinite decimal expansion. But if you take another representation, whereby you define the magnitudes of the sides of the triangle with the square of the lengths, which he calls quadrances, then this problem of infinite representation disappears. From this he got the false belief, that you could eliminate the need of any infinite expansion within plane geometry, in which he was clearly mistaken. Mistaken, because dividing angles in halves leads to square roots in algebra. Therefore if you have a theorem of geometry that requires that angles have to be divided in half two times, the square roots appear again, even in his Rational Geometry.
Still, his attempt is in the right direction. His idea is: if you encounter an infinity in one representation, maybe you can change the representation and then eliminate the infinity. But do not conclude that whenever an infinity appears, then do not conclude that what the infinity tries to express, does not exist. For if you do, you take your own symbolic approach as having an infinite capacity to solve problems. And that causes its own blindness.
As far as Zenos paradox is concerned, I am sorry, but I could not disagree with you more. Here is why.
Zeno formulated his paradox in the form of potential infinity (endless process) not actual. So, since his paradox does not stand the test, this sort of proves that potential infinity does not exist in the real world. Maybe because the space and time are quantized (so far my favorite hypothesis) or for some other reason. And if potential infinity does not exist also actual infinity does not.
As I am able to see it, it is just that simple. But, of course I am very much opened to other interpretations.
It is not ‘that simple’. What you state is TOO simple.
The concept of “potential” was invented by Aristotle, as a reaction to the philosophy of Parmenides, who stated that Everything that is, is. And ‘Nothing can come from nothing”.
Aristotle worked from the premise, that biology, not physics, is the most fundamental science. As such he looked at an egg, that became a chicken. He asked himself: If Parmenides is right, then how can an egg become a chicken? After all, nothing can come from nothing. And Everything that is, is. Therefore, according to Parmenides, either an egg IS a chicken, or it is not possible for an egg to become a chicken, because an egg IS an egg, and therefore cannot be anything else.
This is essentially your argument. Only for Aristotle it was a question, while for you it is an answer.
In any case, to solve this problem, Aristotle came up with the concept of Potential. Armed with this concept, he was able to answer his own question. An egg, as an egg, IS an egg. But it is also a POTENTIAL chicken. Therefore, the chicken already exists in the egg, but not as an actual chicken, but as a POTENTIAL chicken.
The purpose of the whole concept of ‘potential’ is to solve many problems that are caused by just thinking in terms of EITHER something exists, or it does NOT.
The beauty of the concept of Potential is, that it gives a third option. You have: non-existence, potential existence, and actual existence. So, according to Aristotle, two-valued logic is too limited to understand the world. You need three: one for non-existence, the 0, one for potential existence, something between 0 and 1, and actual existence, the 1.
Armed with the concept of Potentiality, it can be understood that there are three forms of mathematics. Mathematics consisting of understandings that tell us that something is impossible. Such understandings usually arises out of assuming that something exists, and then to derive a contradiction from it.
Then there is ‘existence – mathematics’, that is, asserting the existence of something, and then showing a construction leading to the thing that is asserted to exist.
And then there is potential mathematics. That is, not showing a construction, but also showing that the assumption of the existence of that what you imagine does not lead to a contradiction. Therefore, although you have not demonstrated the existence through construction, there is also, as far as you know, no REASON WHY it cannot exist. In this last case you can talk about potential existence.
I also want to point out something I have said before. From this assumption we can begin to think, and then arrive at the conclusion that Zeno has made an implicit assumption which makes that his argument appeared to be plausible, but which was wrong. (Because it was refuted by reality) This implicit assumption is that if we add an infinite number of numbers, no matter how small, and no matter how fast they become smaller, this sum has to be infinite. But we can simply SEE that Archilles WILL OVERTAKE THE TORTOISE. And therefore we can CONCLUDE that this assumption is wrong.
Maybe my English was not good enough. What I meant was: “Zeno has made an implicit assumption which makes that his argument appeared to be plausible, but which was wrong. (Because it was refuted by reality.) This implicit assumption is that if we form the sum of an infinite number of numbers, no matter how small they are, and no matter how fast they become smaller, this sum has to be infinite. But we can simply SEE that Archilles WILL OVERTAKE THE TORTOISE. And therefore we can CONCLUDE that this assumption is wrong.
Moreover, I do not think that your statement about reality being fundamentally discrete helps. One of the great things about quantum mechanics is that it shows how discrete things arise out of resonances of continuous waves. In other words, with the concept of waves and resonances we can see how the discrete arises out of the continuous. Moreover, I think that the whole reason why physics is so preoccupied with string theory is that it does not really express that reality consists of actual vibrating strings, but of vibrations as such, without any underlying substance. So more and more existence appears not to consist of ‘things that vibrate’, but it consists of ‘vibrations that ‘thing’.
The problem with string theory is, I think, that people are so embedded in the ‘is-ness’ of Parmenide et al, that they have real difficulty to transcend the substance – paradigm, and to learn to think in terms of processes alone.
That this is plausible, follows from several very fundamental equations. You have the well-known formula of Einstein, E = m c^2. This asserts, in philosophical terms, that all substance (m) ‘is’ energy. But you also have E = hf, which asserts, that all energy is vibration. Put them together, and you get that all substances ‘are’ vibrations. In other words, the world does not consist of ‘things that vibrate’, i.,e., strings, but of ‘vibrations that ‘thing”.
All quantizations are therefore resonances. And that means, that the world is fundamentally continuous.
I am new to this discussion. I encountered Norman’s videos on YouTube, and found them interesting. I am reading a book about the idea of infinitesimals and the opposition of the religious establishment to the idea of infinitely small in the 17th century. I think Norman’s opposition to infinity, and his math of 21st century (his claim) is really a 21st century version of a 17th century idea. I am speculating here, and I like to know what others think. Here I pasted few passages from the book Infinitesimals, and a paragraph from Norman’s “mild” support of the Intelligent Design. It seems like Norman is opposing infinity because we can not visualize it, and ,at the same time, he promotes the existence of an intelligent designer of the universe because there are a number of mathematical patterns in the world!
Here are the passages:
…“Known as the law of falling bodies, it is one of the first things any student learns today in a high school physics class, but in its time, it was nothing short of revolutionary . It was the first quantitative mathematical description of motion in modern science, and it laid the foundations for the modern field of mechanics— and, in effect, modern physics. Galileo was well aware of the importance of the law, and he included it in two of his most popular works, the Dialogue of 1632 and the Discourses of 1638. Although it relies mostly on Euclidean geometrical relations, it does show Galileo’s willingness to assume that a line is composed of an infinite number of points. That was precisely the question posed to him by Cavalieri in 1621, and whatever answer Galileo gave him, the young monk was not discouraged. During the 1620s he took the idea of the infinitely small and turned it into a powerful mathematical tool that he called the method of indivisibles. The name stuck.”
…”In the coming years, mathematicians who disliked Cavalieri’s method, as did the Jesuits Paul Guldin and André Tacquet, denounced him for his violation of the traditional canons; those who welcomed his approach, as the Italian Evangelista Torricelli and the Englishman John Wallis did, claimed to be his followers while freely making use of infinitesimals with complete disregard for the Jesuat’s carefully thought-out constraints. No one, but truly no one, actually followed Cavalieri’s restrictive system. Cavalieri’s name and his books were often cited by mathematicians when they came under attack by critics of infinitesimals. The heavy and unwieldy volumes, with their contorted Latin, Euclidean structure, and air of solemn authority, provided some cover to later adherents of infinitesimal methods. They thought it was safe to point to the Jesuat master as the source of their system, and the one who had resolved all its difficulties in his learned volumes. After all, as they knew well, hardly anyone actually read Cavalieri’s books.”
…When Jesuits spoke of mathematics, they meant Euclidean geometry. For, as Father Clavius had taught, Euclidean geometry was the embodiment of order. Its demonstrations begin with universal self-evident assumptions, and then proceed step by logical step to describe fixed and necessary relations between geometrical objects: the sum of the angles in a triangle is always equal to two right angles; the sum of the squares of the two shorter sides of a right-angled triangle is equal to the square of the long side ; and so on. These relations are absolute, and cannot be denied by any rational being. And so, beginning with Clavius and for the next two hundred years, geometry formed the core of Jesuit mathematical practice. Even in the eighteenth century, when the direction of higher mathematics turned decisively away from geometry and toward the newer fields of algebra and analysis, Jesuit mathematicians held firm to their geometrical practice. It was the unmistakable hallmark of the Jesuit mathematical school. If only theology and other fields of knowledge could replicate the certainty of Euclidean geometry, they believed, then surely all strife would be at an end. The Reformation and all the chaos and subversion that flowed from it would never have taken root in such a world. This vision of eternal order was, to the Jesuits, the only reason mathematics should be studied at all.”
…If infinitesimals were to prevail, it seemed to the Jesuits, the eternal and unchallengeable edifice of Euclidean geometry would be replaced by a veritable tower of Babel, a place of strife and discord built on teetering foundations, likely to topple at any moment. If Euclidean geometry was, for Clavius, the foundation of universal hierarchy and order , then the new mathematics was the exact opposite, undermining the very possibility of universal order, leading to subversion and strife. Tacquet was not exaggerating when he wrote that in the struggle between geometry and indivisibles, one must destroy the other or “must itself be destroyed.” And so the Jesuits proceeded to do just that.”
From: Alexander, Amir (2014-04-08). Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World (p. 120). Farrar, Straus and Giroux. Kindle Edition
A mathematical perspective
“A question/challenge for those who support the hypothesis of a universe without design: how many interesting mathematical facts do you know? If the answer is: very few, then perhaps your world view has a large slice missing—an aspect that might be even more relevant to the question of cosmic intelligence than the history of the solar system, the fossil record on planet earth, or the genetic sequence contained in our chromosomes. The ancient Greeks, building on earlier Egyptian and Babylonian teachings, initiated logical mathematical reasoning more than 2500 years ago. Thousands of mathematicians working for three millennia have uncovered a vast and intricate, if largely secret, store of mathematical knowledge that compares favourably with the much more widely publicized achievements of those more recent upstarts: biology, chemistry and physics. During this quest, mathematicians have often asked questions such as: How is the remarkable logical coherence of mathematics possible? Why does so much structure and beauty lie at the heart of a world which otherwise often seems chaotic and ugly? Why is mathematics so useful in understanding the physical and chemical laws of the world?”
From: Evolution versus Intelligent Design: a mathematician’s view by N J Wildberger
Interesting historical notes, and I also just read the article written by Norm, saving the math for later (I likely will have to think about the math). I dont accept hard determinism on the grounds that the “thing” that senses, cares, and has concerns actually does make choices and influence behavior. Upon my judment hard determinism leads to absurdities, such as, I know the world works like `X’, but I have to pretend it works like `Y’ in order to function as a human being, which doesnt seem scientific to me. Or in other words, my thinking is that there must be something about `Y’ that is fundamentally important to this even bigger thing `Z’, which contains both `X’ and `Y’.
I also think that even in the rational number system, the numbers themselves are separated by non-finitly small values (smaller than any finite value), insofar as there are always rational numbers between any two rational numbers. It seems that if you do want to set up a real number system foundationally, you are in a sense dealing with rules that coincide with patterns that continuously output rational numbers, and these patterns and rules are the irrational numbers. But does it make sense to call these numbers, or is it a distraction from the objects of consideration, such as the patterns of interest, and rules for generating the rationals such that they conform to the pattern, and any structure that allows for the logical manipulation of the rules such that computation can be completed within a finite number of steps.