We are supposed to have a very clear idea about the `laws of logic’. For example, if all men are mortal, and Socrates is a man, then Socrates is mortal.
Are there in fact such things as the “laws of logic”? While we can all agree that certain rules of inference, like the example above, are reasonably evident, there are a whole lot of more ambiguous situations where clear logical rules are hard to come by, and things amount more to clever arguments, weight of public opinion and the authority of people involved.
It is not dissimilar to the situation with moral codes, where we can all agree that certain rules are self-evident in abstract ideal situations, but when we look at real-life examples, we often are faced with moral dilemmas characterized by ambiguity rather than certainty. One should not kill. Okay, fair enough. But what about when someone threatens one’s loved ones? What moral law guides us as to when we ought to flip from passivity to aggression?
Similar kinds of logical ambiguities surface all the time in mathematics with the modern reliance on axioms, limits, infinite processes, real numbers etc.
Let’s consider here the situation with “infinity”. Most modern pure mathematicians believe, following Bolzano, Cantor and Dedekind, that this is a well-defined concept, and indeed that it rightfully plays a major role in advanced mathematics. I, on the other hand, claim that it is a highly dubious notion; in fact not properly defined; unsupported by explicit examples; the source of innumerable controversies, paradoxes and indeed outright errors; and that mathematics can happily do entirely without it. So we have a major difference of opinion. I can give plenty of reasons and evidence, and have done so, to support my position. By what rules of logic is someone going to convince me of the errors of my ways?
Appeals to authority? That won’t wash. A poll to decide things democratically? No, I will not accept public opinion over clear thinking.
Perhaps they could invoke the Axiom of Infinity from the ZFC axiomfest! According to Wikipedia this Axiom is:
In other words, more or less: an infinite set exists. But I am just going to laugh at that. This is supposed to be mathematics, not some adolescent attempt to create god-like structures by stringing words, or symbols, together.
As a counter to such nonsense, I would like to propose my own new logical principle. It is simple and sweet:
Don’t pretend that you can do something that you can’t.
This principle asks us essentially to be honest. To not get carried away with flights of fancy. To keep our feet firmly planted in reality.
According to this principle, the following questions are invalid logically:
If you could jump to the moon, then would it hurt when you landed?
If you could live forever, what would be your greatest hope?
If you could add up all the natural numbers 1+2+3+4+…, what would you get?
As a consequence of my new logical principle, we are no longer allowed to entertain the possibility of “doing an infinite number of things”. No “adding up an infinite number of numbers”. No creating data structures by “inserting an infinite number” of objects. No “letting time go to infinity and seeing what happens”.
Instead, we might add up 10^6 numbers, or insert a trillion objects into a data set, or let time equal t=883,244,536,000. In my logical universe, computations finish. Statements are supported by explicit, complete, examples. The results of arithmetical operations are concrete numbers that everyone can look at in their entirety. Mathematical statements and equations do not trail “off to infinity” or “converge somewhere beyond the horizon”, or invoke mystical aspects of the physical universe that may or may not exist.
In my view, mathematics ought to be supported by computations that can be made on our computers.
As a consequence of my way of thinking, the following is also a logically invalid question:
If you could add up all the rational numbers 1/1+1/2+1/3+1/4+…, what would you get?
It is nonsense because you cannot add up all those numbers. And why can you not do that? It is not because the sum grows without bound (admittedly not in such an obvious way as in the previous example), but rather because you cannot do an infinite number of things.
As a consequence of my way of thinking, the following is also a logically invalid question:
If you could add up all the rational numbers 1/1^2+1/2^2+1/3^2+1/4^2+…, what would you get?
And the reason is exactly the same. It is because we cannot perform an infinite number of arithmetical operations.
Now in this case someone may argue: wait Norman – this case is different! Here the sum is “converging” to something (to “pi^2/6” according to Euler). But my response is: no, the sum does not make sense, because the actual act of adding up an infinite number of terms, even if the partial sums seems to be heading somewhere, is not something that we can do.
And this is not just a dogmatic or religious position on my part. It is an observation about the world in which we live in. You can try it for yourself. To give you a head start, here is the sum of the first one hundred terms of the above series:
(1589508694133037873 112297928517553859702383498543709859 889432834803818131 090369901)/(972186144434381030589657976 672623144161975583 995746241782720354705517986165248000)
Please have a go, by adding more and more terms of the series: the next one is 1/101^2. You will find that no matter how much determination, computing power and time you have, you will not be able to add up all those numbers. Try it, and see! And the idea that you can do this in a decimal system will very likely become increasingly dubious to you as you proceed. There is only one way to sum this series, and that is using rational number arithmetic, and that only up to a certain point. You can’t escape the framework of rational number arithmetic in which the question is given. Try it, and see if what I say is true!
There are many further consequences of this principle, and we will be exploring some of them in future blog entries. Clearly this new logical law ought to have a name. Let’s call it the law of (logical) honesty. Here it is again:
Don’t pretend that you can do something that you can’t.
As Socrates might have said, it’s just simple logic.
You asked “If you could add up all the rational numbers 1/1+1/2+1/3+1/4+…, what would you get?” and regarding the question stated that “It is nonsense because
you cannot add up all those numbers.” If one was asked to say something interesting about summing 1/1 + 1/2 + 1/4 + 1/8 …, do you think that saying “2 is
the smallest rational number that is greater than the partial sum at any stage in the process.” is logically valid/honest? Also, what about the more succinct
phrase “the value of the sum approaches 2”?
I would agree that a statement like; “2 is the smallest rational number that is greater than the partial sum at any stage in the process” is logically valid, or can be made so. As for “the value of the sum approaches 2” that is a somewhat more problematic: one is flagging here that the meaning of the expression 1/1 + 1/2 + 1/4 + 1/8 … is going to be defined in a certain way. I am not saying that cannot be done, but one has to be very very careful not to build into the definition the assumption that one is “going to do an infinite number of things”.
‘I would agree that a statement like; “2 is the smallest rational number that is greater than the partial sum at any stage in the process” ‘
Isn’t that equivalent saying to limit of the partial sums of sequence equal 2, which is what is meant when people say it equals 2.
When it comes to the logical foundations of mathematics, does it matter what words or symbols we use as long as the meaning remains clear.
In programming we have function overloading, which is where two different functions are given the name, but are differentiated by the type of the objects that are inputted.
When you say 1+1/2+1/4+… =2, you’re using a overloaded definition of the equals sign, it now has a different definition when one input is a sequence and the other is rational number.
I see no logical problem with that.
“2 is the smallest rational number that is greater than the partial sum at any stage in the process.”
The problem with this statement is that it assumes that the notion of “smallest” is applicable to such a comparison.
To better illustrate the problem, consider the following statement:
“2 is the SECOND smallest rational number that is greater than the partial sum at any stage in the process”
How do we know which statement is the true one? The only way to differentiate them is to actually know the numbers involved so that their difference can be resolved because let’s face it, the value of the sum always “approaches the SECOND smallest number next to 2” as well. So how do we know that 2 is not actually the second smallest number and that there is not a nearest neighbor less than 2 that actually best satisfies the condition of “smallest number greater than the partial sum at any stage in the process?”
Even the concrete numbers may be less explicit than expected. After all, there are many possible representations of a concrete natural number. You might write it down in unary notation, in binary notation, or in the commonly used decimal system.
I wanted a more canonical notation system instead, and wanted it to also support rational numbers (and maybe modular arithmetic and q-adic numbers). This gave essentially gave me a notation as acyclic graphs, but deciding whether two numbers are equal in this system leads to questions related to probabilistic identity testing.
For me, this shows that even explicit representations of concrete numbers quickly lead back to questions about the “law of excluded middle”. For the intuitionistic logic started by Brouwer, we have “learned” since how to refute the “law of excluded middle” (in that context). For the ultrafinitist position you encourage, we strongly suspect that the “law of excluded middle” also fails, but being able to prove this would basically be equivalent to resolve some famous open problems in computational complexity.
I like the way your mind works Professor.
I’ve been working again on my GUT (Grand Unified Theory) . I needed to revisit it after a break through in my Quantum Computer Design in which I then had to start to think how to program it, and I reckon I can get around the Infinity problem by putting it off to live in the multiverse.
Our universe is finite (a tiny island within the multiverse) and all our laws of Geometry underpinning the pure and applied math is our beloved UHG it spans the space time limits between the Planck Length and the Planck Second and their inverse’s. A quick napkin calculation will show that this is enough enough time and space to envelope the current estimated size and age of the Universe.
I do all my Geometry in 3D CAD and theorizing in Excel and Wolfram, when I watched your mind blowing video’s on the duality of the Pole and Polar I had that light bulb moment when I saw the point as the end view of a line and a line as the plan view of a point.
If loose the theory of limits that underpins calculus and say any quadrance less than the Planck Length or bigger than it’s inverse is outside our Laws of Physics and in the realm of the multiverse.
I am a string theory devotee and believe all number are just wave counts with differing values of harmonics.
I got an idea for proving the Riemann Hypothesis based on building the case for a finite universe, all number is wave count and Prime Numbers are the group of numbers that have the least harmonics which is what the destructive interference black spots are, in an interference pattern of any multiple wave source.
Musing about it and actually writing a proof that has to be published in a maths Journal for two years before The Clay institute will part with the Million Dollars is a different thing but, for a mad inventor.
I’ll be buying your book shortly I just need to repatriate some light coin cryptocurrency to Aussie Dollar before the Bitcoin halving comes into effect.
Love your ideas keep up the good work.
“Don’t pretend that you can do something that you can’t.”
Beautiful! I love it. I often use phrases like, “Don’t pretend to know what you don’t really know,” but yours actually encompasses that as a special case.
Glad to see you have a blog. I look forward to checking it out. 🙂
Adding up a large enough number of numbers is impossible for a person who lives 150 years. Does that mean it has no definition?
Is any finite task “something you can do”?
I have a suggestion inspired by computer way of processing. What if some where we posit in the axiom of infinite you mentioned that there exist y1 and y2, such that y2<y1 or equal y1 and S(y1)=y2 , this implied a loop somewhere and you can have infinite operation but on a finite set. It 's what happens in computer with overflow number. if you add one each time it will happen that your number overflows and restarts to null, S(n_overflow)=0. Many physical systems do the same, because they are finite they must cycle somewhere.
I do have a simple question, and I apologise beforehand if it is a stupid one.
How does the concept of cardinality adapt to this idea.
Say, for example, the cardinality of the unit interval [0,1].
We are shown it has the same cardinality as the set of all real numbers, which is infinity.
Now, I do understand that you can’t really “get” to infinity, but if you took any set of numbers contained between 0 and 1, you could easily add more numbers, by taking every number between the ones you already have.
How do you define this concept, the idea of always being able to find a new number between any two given real numbers?
Thanks in advance for your answer.
Hi Alexander, Real numbers are not legitimate mathematical objects, despite what people try to tell you. So your question, as posed, doesn’t really make sense. You similarly can’t “take” any set of numbers contained between 0 and 1. It helps if you replace the abstract entities with specific ones. Then things become more clear.
This is ridiculous, every mathematician interprets the statement “adding up an infinite amount of numbers” to mean checking what the limit is, according to a completely rigorous logical definition. You’re arguing with a strawman.
As indeed anyone who has taken a foundations class in an undergraduate mathematics course should know.
Here is an example: consider the sum from k=1 to infinity of 1/(k!)^3. You will agree that this is exactly adding up an infinite amount of numbers. Please tell me what we get. If you can do this by “checking what the limit is” then your point is made.
Actually, there is an answer, obtainable by computer:
k=1 to infinity of 1/(k!)^3 ~ 1.1297.
Norman, I am definitely a fan of your work, and I wish more people knew about it. As this may be the only time I ever communicate with you, I want to take a moment to thank you for the enormous volume of work you’ve presented here and on YouTube, free to the public. I’ve found it personally very enriching to immerse myself in the rich world of your mathematical thinking.
But for all that I think you supply coherent arguments against your detractors, I also think that your position doesn’t achieve what it attempts. Because if you want to say that infinity somehow “doesn’t exist,” then you need time itself to end eventually. You need space to end. And you even make this explicit in your video…
…asking not only about infinite time, but about an object of infinite heaviness (I presume by “heaviness” you mean something with mass = infinity) and claim it makes no sense. But it makes perfect sense. The universe is something, and the universe contains matter. So unless you can tell us that the universe has an end, or that all the matter in the universe exists in the part of it which we can detect, I don’t believe the universe has finite mass.
This is exactly the problem. If we discard infinity, then we are left either with a mathematics that is divorced from reality, or with a universe that must end somewhere. Where? You may like concrete things, but do you envision an enormous concrete wall existing somewhere out there to hold the universe in?
I think what I love about your argumentation is the way in which you won’t be browbeaten by people pretending you don’t understand them, and absolutely won’t go along with absurdity. I, too, will not go along with absurdity. I agree with you that many of Cantor’s arguments fall down; I agree that an axiomatic mathematics which floats free of reality is silly; and I agree that we should balk when people try to tell us things exist which clearly do not. But if you say that “no end” doesn’t exist, what you mean to say is “an end” does exist. We may *imagine* an end to space, or time, or matter, or even the number line! But if you want to assert that there really is no infinity, then it is not unreasonable for me to request that you please show us the last moment of time, or the wall where the universe ends.
Hi Mark, Thanks for your nice comment. I think a good orientation to these problems is to not priorly assume we know what the word “infinite” even means. If someone uses the phrase, ask first — what do you mean by that? If they say something like “it goes on forever”, then ask what is “forever”? — is this just a temporal equivalent to the spacial infinity? This will generate lots of interesting philosophical discussion. If we get tired of that, we can retreat to a more computational position: let’s just stick to talking about what we can write down explicitly. If you can write it down, my computer can potentially deal with it. If you can’t write if down in some form or another, my non-philosophical computer has no idea what you are talking about.
In this way, philosophical questions about “existence” are replaced by computational questions about what can be written down. Less fun perhaps, but maybe more sensible and ultimately useful.
I’m pretty sure that (in suitable predicative constructive but infinitary foundations, optionally with computability axioms) that we can ask that question and get a sensible answer in a way that morally avoids “completed infinities”.
Specifically, we can view real numbers as idealised locations or quantities. To say “the real number which is the infinite sum a0 + a1 + a2 + …” means “given rational numbers p < q, we can show that every sufficiently long _finite_ sum a0 + a1 + … + an is either less than q or greater than p, and take the result of this to be our real number". You can, if you would like, envision a computer program which takes two rational numbers p<q and tells you an n and one of "p", "q" such that all partial sums of length
So to say "the infinite sum 1^-2 + 2^-2 + 3^-2 + … = pi^2/6" means "the results of the program T [which, given rational numbers p<q returns either ("p<",n,evidence) or ("
Similarly, "the sum from k=1 to infinity of 1/(k!)^3" represents an idealised quantity of this form. We can represent it and "real numbers" in general with a program (encodable as a natural number) that compares it to rational numbers, and operations like "addition of natural numbers" are just programs that take (codes for) programs for real numbers, and produce (codes for) programs for real numbers.
This depends on fundamentally constructive/computational interpretation of "for all natural numbers"; predicative constructive mathematics is broadly consistent with a want to avoid completed infinities due to computational interpretations (like the above which, concretely, is most closely related to the category of numbered sets (the category of PERs over Kleene's first PCA)). Other constructive foundations are available.
“And this is not just a dogmatic or religious position on my part.”
Yes, it really is. It is reflected in your language – only things without infinities are ‘pure’, everyone else is illogical and wrong – but not you.
And this is not just a dogmatic or religious position on my part.
Yes – it really is.
I’m not a mathematician, nor a philosopher. I have been puzzled by the concept of infinity as well. I have also been frustrated by the fact that someone people seem to tangibly understand what it is while I never got to that point. So, subconsciously, I liked your stance immediately as a professional mathematician denying this concept altogether.
Obviously I’m not sure if this is correct or not. But I love your new law of logical honesty 100%. Coincidentally, I have been thinking about the laws of logical recently and asked some very similar questions. It’s again just a belief-I’m sure inevitably mathematics will have to meet philosophy somewhere (I just also started reading Kant’s Critique of Pure Reason).
I became very very excited when I became aware of your fabulous work and generous contribution to the general public including myself. I would like to say thank you.