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Has anyone read this article?
This accomplished mathematician gives his opinion on why he doesn't think infinite sets exist, and claims that axioms are nonsense. I don't disagree with his arguments, but with my limited knowledge of axiomatic set theory and logic, I am unable to take sides. Would someone be so kind as to enlighten me on why his arguments are/aren't correct? Thanks
I stopped reading the article at this point:
(6. Axiom of Infinity: There exists an infinite set.
....
And Axiom 6: There is an infinite set!? How in heavens did this one sneak in here? One of the whole points of Russell’s critique is that one must be extremely careful about what the words ‘infinite set’ denote. One might as well declare that: There is an all-seeing Leprechaun! or There is an unstoppable mouse!
Quite frankly, he is using an layperson's interpretation of the axiom and then critiquing this interpretation for being imprecise, when the entire point having these interpretations is to give the gist without being too technical. The common form of the Axiom of Infinity used today is the following (put into words instead of logical symbols):
There is a set $X$ having the property that $\varnothing$ is an element of $X$, and whenever $x$ is an element of $X$, then $x \cup \{ x \}$ is also an element of $X$.
This is a very precise formulation which one can show yields a set which is not finite (hence infinite):
You see that these elements of $X$ get larger and larger without (finite) bound, and so it stands to reason that such an $X$ must be infinite.
Mathematics is a mind game. It doesn't have to do with the physical world. Much like there is no number which is $\frac12$, and there is no number which is $2^{2^{10000}}$, and there is certainly no $\Bbb R^{666}$.
But mathematics is a mind game, where we pretend that for the sake of argument certain objects exists and the axioms are used to describe their properties. In our mind game we agree on certain inference rules, and we try to deduce more properties of these objects using our inference rules and our initial assumptions which we called axioms.
If we take it graciously, the paper is intended to be a tongue-in-cheek essay. There are numerous claims that, if taken at face value, are extremely difficult to defend. Some examples:
On page 6, the author asks, "Do modern texts on set theory bend over backwards to say precisely what is and what is not an infinite set?". Of course they do, it is a simple definition in every text: a set is finite if it can be put in bijection with a natural number, and is infinite otherwise.
At the bottom of page 7, the author claims that the choice of postulates does not arise in his field, which is possible. But, for example, the Whitehead problem in group theory is known to be independent of ZFC, so that proving or disproving it requires more axioms than are generally accepted in mathematics. The Whitehead problem arose first in the context of group theory - not foundations - and only later was proved independent of ZFC.
Near the top of page 9, the author (intentionally?) confuses the property of a mathematical statement being true or false with our ability to prove it is true or false.
The existence of uncomputable reals, which the author discusses on page 11, is well known by results in computability theory to be necessary for statements such as "every bounded increasing sequence of rational numbers converges" to be true - even when we require the sequences themselves to be computable. In particular, the claim on page 12 that the computable real numbers are complete is not constructively provable, as it is disprovable in ZFC.
There are well-written and cogent explanations of different philosophies of mathematics, such as finitism and intuitionism, which the author describes only obliquely. This paper might be better as something to read after you are familiar with those philosophies, so that you get the jokes that the author is making.
Recently I was reminded of the following gem of an aphorism: "The most annoying thing about an incorrect proof of a correct theorem is that it is very difficult to give a counterexample." It is certainly true that infinite sets do not necessarily "exist" in most uses of the word other than the mathematical one. It is not, however, true that accepting set theory as foundations foces one to believe in such existence in any sense beyond the mathematical. Furthermore, the existence of "infinite sets" is no more contentious than the existence of "finite sets", in my opinion.
I am not a logician (yet), but the picture in my head is as follows. Mathematicians at the end of the day deal with certain systems of rules on how to manipulate symbols on a piece of paper. Such systems are composed of two parts: a language which consists of the rules that say which strings of symbols are valid (i.e. are sentences or formulas), and the transformation (inference) rules which say how to transform certain (collections of) sentences and formulas into other sentences and formulas. Formally, this is all we do as mathematicians: we come up with languages and inference rules, pick some sentences or formulas in the language that seem interesting and then we go on and try to obtain certain other interesting sentences and formulas (you get at mathematical logic if you ask yourself whether you can obtain certain interesting sentences and formulas at all).
From this formal perspective, the relation to the real world is that occasionally a more scientifically inclined mathematician (or more commonly, a mathematically inclined scientist) would use or create a language in which to describe the things in the world he or she observe, and the relationships between the things he or she hypothesizes. Then, they apply whatever set of inference rules they use (usually basic logic) to their initial conditions and laws, and thus arrive at a new sentence or formula, which they label a prediction about the real world. Then they go and see if the prediction is true. If yes, they say that the formal system they came up with describes the real world, which is never true: the formal system only models the real world, i.e. functions to predict rather than describe things about the real world.
Things like the natural numbers, basic rules of arithmetic, or the finite set theory Wildberger prefers, are simply formal systems which have always given correct (when testable) predictions about the real world. What people actually mean when they say that 1+1=2 is a self-evident statement is that in almost all contexts, the statement "one thing and another thing give us two things" has proven true. But this is of course tautological, since the idea of 1+1=2, i.e. the language of arithmetic and its basic properties are considered interesting exactly because of the fact that they model so many phenomena that we observe extremely well. It is absurd, however, to claim that the number 1 "exists" in any sense other than the mathematical, which is that there is a certain practice we engage in, which has always accurately predicted certain situations in the real world (i.e. if I take one apple, and another apple, I now have two apples).
What about "infinite sets" and ZF(C)? What aspect of reality do they model? Well, ZFC models the very real practice of doing mathematics in the above sense. It gives symbols and rules with which to express strings of symbols (the set of all finite strings), the language (the subset of all valid strings), and inference rules (functions on sets of valid formulas). We even have for certain kinds of formal systems Godel's completeness theorem which states that if a theory is consistent (its set of theorems/formulas derived from axioms does not include "P and not P" for any P), then ZFC can model that theory in a standard way. Assuming that ZFC is consistent, the implication goes the other way as well, i.e. ZFC models only consistent theories if it is itself consistent.
For this reason, almost all mathematicians of an object (in a theory) have agreed to understand mathematical existence to mean that any way in which ZFC models that theory, the object is represented in the model. This is why defining, say, the rational numbers or the real numbers as equivalence classes of whatever is not as insane as it might seem: it is actually showing that the rationals and the reals exist in the sense that their theories pass the test of consistency relative to ZFC. This is important if we want to have some standard by which to be confident that these formal systems (of the rational numbers, of the real numbers) are free of contradictions, i.e. would not simultaneously predict "P and not P". Otherwise, because of how our inference rules are set-up, their theorems are trivial (every formula is a theorem), and thus their utility as models of the real world is null.
This joker is just playing to the gallery. "Maths $-$ who needs it? Ha ha ha!"
To take a specific example, on page 10 he ridicules the standard definition of a rational number as an equivalence class of ordered pairs of integers. As I hope you know, this is perfectly standard, and no "accomplished mathematician" should have any problem with it at all.
Wildberger makes the point
clear definitions are necessary
and
People use the term ‘Axiom’ when often they really mean definition. Thus the ‘axioms’ of group theory are in fact just definitions. We say exactly what we mean by a group, that’s all. There are no assumptions anywhere.
Oddly, he seems to miss the obvious application of this idea in regards to universes of sets. I suppose it shouldn't be that surprising, due to all the effort he spends setting up a caricature of modern set theory so that he could proceed to mock the straw-man.
Nor does he seem to take the time to apply this principle to whatever philosophy he is suggesting. Read through his article; is there anything in there you could have predicted that he would say before he actually said it? A fair bit, probably, but very few (if any) of those predictions will be because due to the correct use of logic as applied to clear definitions.
Let's consider one particular point. He talks about the natural numbers. This is clearly a concept he accepts in some fashion, even if he dislikes the modern treatment.
(aside: much of the controversy would go away if he would simply do something like define "Wildberger sets" and "the Wildberger numbers" and develop their theory. Instead, he leaves them undefined, calls them "sets" and "natural numbers", and then tries to browbeat everybody to stop using the usual meaning for "set" and "natural number")
He even admits implicitly that "natural numbers" are some sort of object that can be reasoned with -- e.g. one can make statements such as "$f(n) = n^2 + 1$ defines a function that inputs natural numbers and outputs natural numbers" -- so in his treatment, "natural number" is clearly not simply some metaconcept.
Now recall Cantor's approach to set theory. One of the most basic ideas is comprehension: if $P$ is a proposition, then
$$ \{ x \mid P(x) \} $$
is a set. One may have some a priori ideas about sets as some sort of "collection" (but then, what is a "collection"?), but in Cantor's set theory, the notion of set equates to the notion of predicate. And I don't believe the underlying idea was particularly new -- philosophers had been struggling with such things for a long time -- the novel feature is that it was cleanly and precisely stated and one could reason rigorously with it, and that Cantor was willing to fully explore what could be done with it.
So if Wildberger is willing to grant that we can reason about "natural numbers" -- and even go so far as allowing them to be some sort of object -- then the natural numbers are a Cantor set. (of course, it's apparently not a Wildberger set, but I'm not talking about Wildberger sets, we're talking about Cantor sets)
By any reasonable definition of the word "finite", the natural numbers should not be a finite Cantor set. Thus, infinite Cantor sets exist, even in Wildberger's way of thinking. Assuming, of course, that Wildberger's way of thinking can even give a reasonable definition of "finite".
Well, I should be careful; Wildberger has not used clear definitions. If he is talking about the natural numbers as he claims to be, then I can conclude that even in Wildberger's mathematics, infinite Cantor sets exist. However, if he is talking about Wildberger numbers instead, I honestly don't know if they form an infinite Cantor set. e.g. I'm not really sure if there is a largest Wildberger number or not.
Now, mind you, people like to study other universes of sets. The universe of finite sets, for example. This has rather severe limitations, and it's not generally adequate to study mathematics -- e.g. the notion of a "function whose inputs are natural numbers and whose outputs are natural numbers" cannot be encoded in such a universe. We can recover a fragment of such a notion by defining things like Turing machines.
A lot of controversy would have been avoided if Wildberger simply said "I want to study constructive analysis rather than real analysis" and maybe even presented reasons why teaching students constructive analysis would be better than teaching calculus and real analysis. But then, I suppose that would have had the drawbacks of being less provocative, and actually exposing his rationale to be critiqued (if he even has one!).
I totally agree with Asaf Karagila here. Mathematics is a mind game, but Professor Wildberger assumes that mathematics should relate to the real world:
Elementary mathematics needs to be understood in the right way, and the entire subject needs to be rebuilt so that it makes complete sense right from the beginning, without any use of dubious philosophical assumptions about infinite sets or procedures. Show me one fact about the real world (i.e. applied maths, physics, chemistry, biology, economics etc.) that truly requires mathematics involving ‘infinite sets’ ! Mathematics was always really about, and always will be about, finite collections, patterns and algorithms. All those theories, arguments and daydreams involving ‘infinite sets’ need to be recast into a precise finite framework or relegated to philosophy.
But as far as I know, mathematics is not, and should not be about reality. I would like to quote Albert Einstein here
"As far as the laws of mathematics refer to reality, they are not certain, as far as they are certain, they do not refer to reality."
I'll focus on this question: "Does the idea of an infinite set make sense?" The author of the article answers no; in general, mathematicians say yes, but an appropriate answer would be a question: "Where?"
The pattern came up multiple times in the history of mathematics: the number zero, negative numbers, irrationals, complex numbers;
Do negative numbers make sense?
No, if I'm thinking of using numbers to count sheep.
Yes, if I'm thinking, for instance, about temperatures in °C.
Do complex numbers make sense?
No, if I stay on a line and can't see anything but real numbers.
Yes, if I expand my horizon to a plane and assign numbers to its points.
So, in general,
Does ___ make sense?
It depends on where you are.
And thanks to math you can give actual names to the "places" in question: N, Z, C, ZFC...
Each place is characterized by a set of rules, and has its own inhabitants.
Asking if "-1" exists is like asking if "Harry Potter" exists. In which story? "-1" exists in C but not in N, in the same way you can't find an "Harry Potter" in the Lord of the Rings but you can, for instance, in J. K. Rowling's stories.
So, let's ask: Does the idea of an infinite sense make sense? It depends on where you are. If you put yourself in ZFC, it does make sense, and it's not the only place where this happens. But, you can also put yourself in a place where infinite sets do not exist. No one stops you from doing that: you can choose which books to read, which tools to use, which songs to sing. Why use negative numbers if all you want to do is count your ten sheep? (in the same vein, why use zero, or eleven?)
The problem is that the author of the article suggests that no place with infinite sets makes sense, and that's like having ten sheep and claiming "eleven" doesn't make sense, not in your world, not in anyone else's.
That claim only makes sense if directed at contradictory theories (naive set theory, for instance), but you would have to prove the contradictions you claim are present to be credible (for example, by finding something like Russell's paradox).
Without taking a stand one way or another on the author's philosophical and pedagogical claims, I would like to correct him on one small point. When he writes (mockingly)
it surely is possible to dissect a solid unit ball into five pieces, and rearrange them to form a solid ball of radius two.
he seems to be misstating a result of Raphael M. Robinson related to the Banach-Tarski paradox. The correct statement:
It is possible to dissect a solid unit ball into five pieces, and rearrange them to form two solid unit balls.
It is also possible to dissect a solid unit ball into some finite number $N$ of pieces and rearrange them to form a solid ball of radius two, and I don't know what's the smallest $N$ that works, but I'm pretty sure it's bigger than $5.$
If no infinite sets exist, then the natural numbers are not infinite. Consequently, the natural numbers are finite. It follows that there exists a last natural number, call it L. Since it's the last natural number, then S(L) the successor of L or (L+1) does not exist. In other words, at some point, no matter how "high" you count, you will NOT have the ability to count to another number and no one, nor anything in the universe will. Nor will you have the ability to find a natural number greater than L ever. So do you believe that there exists a last natural number? Does the set of natural numbers have a least upper bound? Does the set of natural numbers have a greatest member?
If not, then you reject the notion of infinite sets not existing.
I don't think any mathematical object exists in real world, but there are some accurate defined ideas in mathematics that exists in human mind. Few people would argue against the existence of numbers, lines, circles, sets and so on, but not all definitions are equal convincing.
A set is really an intuitive idea and it is intuitively clear what it means that an element belongs to a set. This intuition is somewhat lost in axiomatic set theory, since the axioms merely are rules for what to be called a set than a definition of a unique relation $\in$.
If an idea has to be convincing in order to be existing, then off course an axiom like It exists an infinite set isn't ideal. Almost any mathematical object prompts for a generalization to infinity, but the process is often rather technical, in order to convince mathematicians. And I would really like to see other kind of models of sets than the usual.
I think that relevant models of infinity has a place in mathematics. Any straight line includes a prompt for some model of infinity. But then, lines exists in reality only as intellectual experiments.
The author argues about very big numbers, to big to be "written in universe". I can agree with him that we tend to be arrogant when thinking about numbers and a "set of all numbers", because almost any number is mindbogglingly greater than any human being ever is going to grasp. But arguments including a limited universe isn't adequate.
I believe in generalizations, as ideas in human mind: good generalizations.
