Here is a quote from the online Encyclopedia Britannica:
The Bohemian mathematician Bernard Bolzano (1781–1848) formulated an argument for the infinitude of the class of all possible thoughts. If T is a thought, let T* stand for the notion “T is a thought.” T and T* are in turn distinct thoughts, so that, starting with any single thought T, one can obtain an endless sequence of possible thoughts: T, T*, T**, T***, and so on. Some view this as evidence that the Absolute is infinite.
Bolzano was one of the founders of modern analysis, and with Cantor and Dedekind, initiated the at-the-time controversial idea that the `infinite’ was not just a way of indirectly speaking about processes that are unbounded, or without end, but actually a concrete object or objects that mathematics could manipulate and build on, in parallel with finite, more traditional objects.
A multitude which is larger than any finite multitude, i.e., a multitude with the property that every finite set [of members of the kind in question] is only a part of it, I will call an infinite multitude. (B. Bolzano)
Accordingly I distinguish an eternal uncreated infinity or absolutum which is due to God and his attributes, and a created infinity or transfinitum, which has to be used wherever in the created nature an actual infinity has to be noticed, for example, with respect to, according to my firm conviction, the actually infinite number of created individuals, in the universe as well as on our earth and, most probably, even in every arbitrarily small extended piece of space. (G. Cantor)
One proof is based on the notion of God. First, from the highest perfection of God, we infer the possibility of the creation of the transfinite, then, from his all-grace and splendor, we infer the necessity that the creation of the transfinite in fact has happened. (G. Cantor)
The numbers are a free creation of human mind. (R. Dedekind )
I hope some of these quotes strike you as little more than religious doggerel. Is this what you, a critical thinking person, really want to buy into??
From the initial set-up by Bolzano, Cantor and Dedekind, the twentieth century has gone on to enshrine the existence of `infinity’ as a fundamental aspect of the mathematical world. Mathematical objects, even simple ones such as lines and circles , are defined in terms of “infinite sets of points”. Fundamental concepts of calculus, such as continuity, the derivative and the integral, rest on the idea of “completing infinite processes” and/or “performing an infinite number of tasks”. Almost all higher and more sophisticated notions from algebraic geometry, differential geometry, algebraic topology, and of course analysis rest on a bedrock foundation of infinite this and infinite that.
This is all religion my friends. It is what we get when we abandon the true path of clarity and precise thinking in order to invoke into existence that which we would like to be true. We want our integrals, infinite sums, infinite products, evaluations of transcendental functions to converge to “real numbers”, and if belief in infinity is what it takes, then that’s what we have collectively agreed to, back somewhere in the 20th century.
What would mathematics be like if we accepted it as it really is? Without wishful thinking, imprecise definitions and reliance on belief systems?
What would pure mathematics be like if it actually lined up with what our computers can do, rather than with what we can talk about?
Let’s take a deep breath, shake away the cobwebs of collective thought, and engage with mathematics as it really is. Down with infinity!
Or somewhat less spectacularly: Up with proper definitions!
Fermat already knew how to get rid of the infinities in calculus, before calculus was invented. Nilpotent infinitesimals. https://plus.google.com/108269652526642085924/posts/EcrwP9cDBz3
Calculus without limits. Automatic differentiation. Adequality. http://arxiv.org/abs/1210.7750
Nice, thanks for those links Tom. There are algebraic, finite, concrete and logical ways of replacing what is currently vague, imprecise and reliant on infinite processes.
https://en.wikipedia.org/wiki/Dual_number is a nice article on the modern treatment of Fermat’s adequality ideas using matrices to form a ring with a nilsquare element that “is” an infinitesimal: they are not “ghosts of departed quantities”, they are 2×2 matrices.
Great article; I completely agree with you.
Regarding proper definitions, terms such as “for n=1 to infinity” and “infinitely many” seem to crop up a lot in definitions.
I need to understand what these things mean in order to understand how real numbers are defined as equivalence classes of Cauchy sequences of rational numbers, Dedekind cuts, or certain infinite “decimal representations”.
When pure mathematicians talk about a summation “for n=1 to infinity”, or when they claim something can occur “ad infinitum” or “infinitely many” times what number type are they referring to (do they even know or think about it themselves)?
It can’t be natural numbers or integers because these cannot “go to infinity”. By definition they can only take finite values. It can’t even be ‘reals’ because these do not allow infinity as a value.
So presumably they must be talking about something like the Hyperreal numbers or the Surreal numbers? Now as far as I am aware, these are extensions to the reals, which they both take as already being well-defined.
So it appears to me that all definitions of ‘reals’ are based on the assumption that ‘reals’ are already well-defined. So real numbers are not well-defined at all, but pure mathematicians still accept them. Why do they do this?
I don’t have a problem with infinity as a concept, a label we give to certain divergent limits. Limits are not necessarily numbers, however, and infinity is certainly not a number. Numbers are things you count with and compute with. As a fundamental property they have a definite magnitude. Infinity has no such definite magnitude. All we can say about it is that it is always bigger than it needs to be, no matter how big that is. We may also be able to compute it’s sign.
I do have a problem with Cantor’s diagonal proof of multiple infinities. Cantor’s proof relies on an absurd nonconstructive definition of a certain real number: that real number between 0 and 1 which is distinct from every other real number between 0 and 1 in at least 1 digit of their decimal expansion after having enumerated all real numbers between 0 and 1 in their entirety and assigning each to a natural number.
The fact that he disguises his nonconstructive definition as a constructive one by specifying it in a pseudo-procedural format is what has confused mainstream mathematicians into buying into his absurd proof. Its fallacy ought to have been apparent from the start. The computation implied by the definition fails at the first attempt to write down all the digits of the first real number in the enumeration. The argument fails to compute, but not at the place where the _ reductio ad absurdum_ argument anticipated. It failed in the absurd setup for the definition of the unique number, which depends on being able to first construct an infinitely long and wide table of digits, not the attempt to find another natural number to pair the diagonally unique number to.
If restricted to a more constructive approach that mandates only finite steps that each require only finite amounts of paper and ink, or equivalently, computer memory bits and computation time, Cantor’s argument is easily seen as fallacious. The real number he defines by his diagonal pseudo-construction does not and cannot exist because it is not constructable, relying as it does on an infinitely long process that can never produce a proper answer. Of course had Cantor succeeded in producing the number he intended, there would be no problem at all incrementing the last natural number given out and providing yet another natural number to pair it to. The natural numbers are, after all, infinite. There are always enough.
What Cantor set out to prove is that there were “more” real numbers between 0 and 1 than there were natural numbers. What he actually proved is that his diagonal number does not exist because it’s definition has no referent.
“Its fallacy ought to have been apparent from the start. The computation implied by the definition fails at the first attempt to write down all the digits of the first real number in the enumeration. The argument fails to compute, but not at the place where the _ reductio ad absurdum_ argument anticipated. It failed in the absurd setup for the definition of the unique number, which depends on being able to first construct an infinitely long and wide table of digits, not the attempt to find another natural number to pair the diagonally unique number to.”
A proof by contradiction (or in this a case a proof of negation which is the constructive version http://math.andrej.com/2010/03/29/proof-of-negation-and-proof-by-contradiction/) is kind of meant to have a fallacy in it, the assumption you make at the start of the proof by contradiction is always going to be wrong, so all proofs by contradiction go wrong at the start, the setup will always be absurd.
When doing the proof constructively you never try to write down the whole table. You assume you have a function f(n,x) which is meant to mean the xth digit of the nth number. You then construct a new function g(x) that uses f, such that there is no i such that g(x) is the same as f(i,x). That’s the diagonal argument, and it’s perfectly constructive because I can create a program that outputs a valid program g when given a program f.
The “constructive proof” does not work either. It assumes that we know what we mean by a “function”. Unfortunately we do not have a good definition of that, and once we start to inquire we see that there are a range of possible meanings. The argument then falls apart in different ways, depending on which definition you adopt. (That is one of the reasons why mainstream mathematicians DON’T like to define precisely what a “function” is!)
What about primitive recursive functions, they are definable because you can always check that a function is primitive recursive.
The proof then shows that you can’t enumerate over all the PR reals using a PR function, e.g. no bijection between naturals and (PR) reals, which is how “different size” is defined. It doesn’t really make sense to call it size anymore when talking about computable sets though.
A very naive question. What areas in mathematics will be left standing without the notion of infinity? Algebraic topology? Manifold theory? Computability theory, e.g. Turing’s Halting Problem?
We will have to change a lot of it, that is certainly true.
Can you give an example of what will have to change?
P.S. All topological spaces would compact, yes?
No, we cannot say, because we have not yet defined a proper theory of “topological spaces”. Once we do that, then we will be in a position to make meaningful statements about them.
What is the problem with the current definitions of a topological space. The open subsets definition seems fine, it doesn’t use infinity anywhere, and you can make finite topological spaces, you can take any finite set and make the discrete space.
If there are only finite sets then all topological spaces would be compact.
I’ll admit it makes them less interesting, but I would bet that if they had anything continuous about them they would have to be finite.
In SIA, smooth infinitesimal analysis, infinities are done away with, i.e. are made unnecessary, by changing the definition of number (much like is done to allow square roots of negative numbers). epsilon, rather than being the ghost of a departed quantity, is simply the unit for another, perpendicular dimension. The logic becomes intuitionistic, and all functions are continuous.
Interesting, Tom, I have not heard of this SIA. What is the best reference?