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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

Andrés E. Caicedo
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Nicolas Bourbaki
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    I doubt you can completely falsify/prove his statements correct. As far as humanity knows, different set theories are uncomparable and subject to belief/specialization, not truth. – CBenni Apr 09 '13 at 19:26
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    From the article - "Mathematics that makes complete sense tends to parallel the real world and be highly relevant to it". But according to Einstein, "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." – Joshua Shane Liberman Apr 09 '13 at 19:31
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    I really like how he emphasizes on page 7 that the "existence of an infinite set" is basically a wrong postulate, and not in any way related to the real mathematics he does, namely representations of Lie Groups...I really wonder how much of his work is about representations of FINITE Lie Groups, and if it is, if anyone cares... – N. S. Apr 09 '13 at 19:44
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    "These grammatical constructions do not create concepts, except perhaps in a literary or poetic sense" As if a concept should correspond to something that exists. – leonbloy Apr 09 '13 at 19:51
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    A paper about any mathematical subject without any list of sources ? I wouldn't, and I won't, waste my time with it. – DonAntonio Apr 09 '13 at 20:15
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    In my opinion an infinite set doesn't need to "exist" any more than the number 3 needs to "exist", they are all imaginary concepts and there is nothing wrong with that for the purposes of doing useful/interesting mathematics – wim Apr 10 '13 at 03:19
  • Not all set theories have all of those axioms. Fuzzy set theory, for instance, doesn't have an axiom of extensionality. – Doug Spoonwood Apr 10 '13 at 04:18
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    @N.S. Dear N.S., The author is not going to claim that the Lie group $G_2$ (as one example, which I am bringing it up because it appears in the text) is finite. His objection is rather to a certain axiomatic treatment of mathematics, including the axiomatic treatment of infinite sets. Regards, – Matt E Apr 18 '13 at 14:59
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    Isn't this discussion best suited for Philosophy? – pamatt Apr 24 '13 at 17:33
  • I'm removing all the off-topic comments from this thread. The original discussion is preserved here. – Willie Wong Jun 19 '13 at 10:30
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    You cant argue with "Axiom: there exist an infinite set" and "ZOMG Axioms cant be wrong!!!!11". That is some very deep mathematics right there. – Grandmaster Jul 07 '13 at 23:40
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    there's a lot about all-seeing leprechauns in that paper. – Rustyn Apr 14 '14 at 06:56
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    @wim But I find $3$ more intuitive than $\infty$ – Amr Apr 19 '14 at 17:14
  • If jumping straight into infinity is too daunting, start with the finite. See my blog posting, "Infinity: The Story So Far" at https://dcproof.wordpress.com/ There I start with a non-numerical, informal development of the notion of a finite set, eventually developing various formal notions of an infinite set. – Dan Christensen May 07 '15 at 17:05

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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):

  • As $\varnothing$ is in $X$, then $\varnothing \cup \{ \varnothing \} = \{ \varnothing \}$ is an element of $X$.
  • As $\{ \varnothing \}$ is in $X$, then $\{ \varnothing \} \cup \{ \{ \varnothing \} \}= \{ \varnothing , \{ \varnothing \} \}$ is in $X$.
  • As $\{ \varnothing , \{ \varnothing \} \}$ is in $X$, then $\{ \varnothing , \{ \varnothing \} \} \cup \{ \{ \varnothing , \{ \varnothing \} \} \} = \{ \varnothing , \{ \varnothing \} , \{ \varnothing , \{ \varnothing \} \} \}$ is in $X$.
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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.

user642796
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    The author doesn't seem to argue that mathematicians can construct numbers like this, but takes issue that this construction requires more materials than exist in the universe, even on an atomic level. I don't need infinite sets to do that, just let $x = $ the number of atoms in the universe $+ 1$. – Joshua Shane Liberman Apr 09 '13 at 20:39
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    It also stands to reason that X doesn't actually exist because the algorithm for generating it does not terminate. We can argue that something doesn't exist until the function which computes it returns to the caller, so to speak. – Kaz Apr 09 '13 at 22:25
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    In functional languages we can use lazy objects to implement such structures. But with that comes the understanding that the structures are virtual, which is a way of saying they don't actually exist. Some of the usual operations are not possible on these structures, because they try to force the structures to a full instantiation. For instance, lists have length, right? But if we take the length of a lazy infinite length, the force will never terminate (or rather, it will run out of memory on a real machine). – Kaz Apr 09 '13 at 23:01
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    You could mention, that Von Neumann proposed that all natural numbers can be bootstrapped out of the empty set by the operation you described. – Christian Ivicevic Apr 09 '13 at 23:08
  • Natural-number-like sets (ones on which Peano's Axioms hold) can be extracted from any Dedekind-infinite set using a only a subset axiom (specification in ZF). Postulating the existence of such a set seems more esthetically pleasing to me than the ZF axiom of infinity. I never did like constructions like ${\phi,{\phi,},{\phi.....$ – Dan Christensen Apr 10 '13 at 17:59
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    @Igor: I'll be generous and dignify your insult of me with a response; don't expect another reply from me. One does not need to read every word of a screed to determine that it is without merit. Just as I don't need to listen to Justin Bieber's entire œuvre to determine that he's a horrible songwriter. The author of the article in question was clearly using a caricature treatment of axiomatic set theory to attack axiomatic set theory. He wasn't willing to attack the practice as it is done, but resorted to what is a straw man argument to conclude that set theory is a vacuous enterprise. – user642796 Apr 18 '13 at 17:08
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    It's not true that the essay is "using a layperson's interpretation and critiquing this interpretation for being imprecise". Reading the whole thing is the easiest way to see that. – zyx Apr 21 '13 at 04:49
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    @zyx: After stating the "Axioms of Zermelo-Fraenkel" Wildberger goes on to single out his statement of the Axiom of Infinity as being imprecise (this is his appeal to Russell which I have quoted above). Perhaps I should proclaim algebraic topology to be stupid because homology groups supposedly "count the number of holes" in a space, but how can there be a hole in something that has no physical existence?! and, besides, a "hole" is a pretty imprecise concept to make mathematical reference to! [cont...] – user642796 Apr 21 '13 at 05:35
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    [...inued] Of course, I know better than this, and that the "count the number of holes" idea is a somewhat loose interpretation of the actual state of affairs. Similarly, the Axiom of Infinity is given a very precise statement, granting the existence of what is called an inductive set. (If you want a completely formal statement of this axiom in the language of set theory, I can give it to you, but I hope this is not necessary.) [cont...] – user642796 Apr 21 '13 at 05:36
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    [...inued] From this we can get a definition of finiteness (being equipotent to an element of the smallest inductive set), and thus infiniteness (being not finite), and then show that any inductive set cannot be finite (using the precise definition given above). Might Wildberger actually be trying to advance a finitist (or possibly even a ultrafinitist) position of mathematics? Perhaps. But he weakens his position incredibly by making false statements about the precision set theorists have with respect to the concepts of finiteness/infiniteness. – user642796 Apr 21 '13 at 05:36
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    @ArthurFischer Wildberger probably takes issue that infinite mathematical sets don't correspond "sufficiently well" to non-finite sets in the physical world. I like to think of a bottle with fluid as a nice model for a "compact" (i.e. quasi finite) physical set, and of a toy ballon with gas as a nice model for a "non-compact" physical set. The problem is, neither finite sets nor infinite sets are a good model for the toy ballon. If we keep filling gas into it, it will explode, but we can't say exactly when. A "potential infinite" is a better model than an "actual infinite" for this fact. – Thomas Klimpel Apr 21 '13 at 14:07
  • [continued] However, the feeling that using "potential infinite" instead of "actual infinite" would create more problems and paradoxes than it's worth is probably correct. But while Hilbert's vision that "actual infinite" could be justified by finitistic methods was disproved by Gödel in a certain sense, the same question for the "potential infinite" never received the same attention. However, the "actual infinite" is also problematic, not because it could be self-contradictory, but because some non-sense theorems will become true (like doubling the sphere), if the axioms are mighty. – Thomas Klimpel Apr 21 '13 at 14:21
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    @Thomas: Again, I take issue with Wildberger's use of a straw-man when criticising set theory. Reading his script it appears that Wildberger doesn't even accept potential infinity, with the basic idea that we have no way to compute basic facts about certain huge natural numbers, and they also have no physical representation, and so in what sense can those numbers be said to exist? [cont...] – user642796 Apr 21 '13 at 15:08
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    [...inued] Note that Banach-Tarski is more a consequence of the Axiom of Choice than Infinity. It is consistent with the usual axioms of set theory minus Choice that all sets of reals are Lebesgue measurable, and therefore Banach-Tarski cannot simply be a consequence of infinite sets. But just because a theorem doesn't match our intuitions well does not make it nonsense. Consider those mathematicians who came ever so close to discovering non-Euclidean geometry, excepting their belief that consequences of negating Euclid's fifth were absurd, as opposed to describing something else. – user642796 Apr 21 '13 at 15:09
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    Wildberger is a professional mathematician, so he knows that he has to accept "some version" of infinity. I'm not sure he intents to use a straw-man, when he writes: "The ‘Axioms’ are first of all unintelligible unless you are already a trained mathematician." And he is criticising set theory mainly in its role as foundation of mathematics. You write "he is using an layperson's interpretation of the axiom", but Wildberger has a point that even the undergraduate student is in this layperson's position. Carl Mummert answer that it's a "tongue-in-cheek essay" is good, I really laughed out loud. – Thomas Klimpel Apr 21 '13 at 16:01
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    @Thomas: I may be a professional barista, but if I will suddenly claim tea leaves to make excellent coffee - certainly this claim will be met with ridicule. Wildberger may be a professional mathematician, but claiming that axioms are imprecise and that mathematical existence must be related to physical existence and therefore "existence of an infinite set" is equivalent to "existence of leprechauns" will be met with the same ridicule as the coffee-brewing tea leaves claim. – Asaf Karagila Apr 21 '13 at 16:53
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    @Thomas: It is not clear that he limits his attack of "infinite set theory" to its possible foundational role. But his statement of the axioms must be intentionally imprecise, and he compounds this by glibly stating "Do modern texts on set theory bend over backwards to say precisely what is and what is not an infinite set? Check it out for yourself – I cannot say that I have found much evidence of such an attitude, and I have looked." Quite frankly, he is either lying (about looking for such precise definitions) or obtuse (for not being able to find any having looked). [cont...] – user642796 Apr 21 '13 at 17:39
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    [..inued] Carl Mummert's answer states "If we take it graciously, the paper is intended to be a tongue-in-cheek essay" [emphasis mine]. Given other materials he has produced, and his naming of one Prof. Dr. Wolfgang Mückenheim (another "professional mathematician" and vocal critic of Cantor's work, whose supposed enumerations of the reals always seem to omit $\frac{1}{3}$) as a supporter, I am much less convinced that we should be so gracious. – user642796 Apr 21 '13 at 17:40
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    If you say his name three times in front of a mirror he will come to haunt you and put you in his Matheology texts! Beware of the M. Also M. is a physicist who insists that he knows mathematics better than all mathematicians - for some reason. – Asaf Karagila Apr 21 '13 at 19:18
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    @Arthur: Until you can prove that the axioms of ZFC are consistent, it seems ungenerous to ridicule someone for not believing them. You are not, I hope, under any illusions that your answer does anything like proving the existence of an infinite set? – Pete L. Clark Apr 21 '13 at 22:32
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    @Pete: I can easily prove you the consistency of the axioms of $\sf ZFC$ from the axioms of the Tarski-Grothendieck theory. We all [read:most] take $\sf PA$'s consistency for granted, what's the difference between that and $\sf ZFC$? – Asaf Karagila Apr 21 '13 at 22:34
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    @Asaf: Most working mathematicians do not "take PA's consistency for granted". Rather (and I'm just being honest, not intentionally harsh) they don't know and don't care what PA is. And assuming that set theory is consistent so that you can go about your business of doing mathematics (even set theory, which is of course part of mathematics) is not at all the same thing as believing in some philosophical sense that set theory is consistent. – Pete L. Clark Apr 21 '13 at 22:40
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    In other words, I claim that "ridiculing non-believers" is a fundamentally non-mathematical thing to do. I am more than happy to use the Axiom of Choice in my work -- I usually do so without comment and, honestly, even thoughts of whether I am using it or not are idle distractions from my actual work -- but let's not draw quasi-religious conclusions from that. – Pete L. Clark Apr 21 '13 at 22:44
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    @Pete: Of course. But the point of my comment was that people just assume that the foundations that were laid by their predecessors are solid. As $\sf ZFC$ is somewhat of a "folklore" foundation, it is the common basis to question; but if you really don't care about foundation then you don't really question it -- and if you do question it, then you know how to question it. Just writing "Oh, their axioms be stoooopeed!" is not a proper way to raise doubts in the foundations of mathematics. – Asaf Karagila Apr 21 '13 at 22:45
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    @Asaf: I can't disagree with any of what you said. Moreover, as I hope you realize, my intention is not to raise any doubts in the foundations of mathematics. Rather, it is to point out that unquestioning belief in the truth of the foundations of mathematics is simply an article of faith. You can have it or not and go about your mathematical business equally well. – Pete L. Clark Apr 21 '13 at 22:48
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    @Arthur: I have now spend considerable time searching the internet to determine whether Wildberger is (considered to be) a crackpot or not. His essay may be provocative and controversial, but the general opinion seems to be that it is unwarranted to compare him to infamous people like Wolfgang Mückenheim or John Gabriel (should I know these?). There is also general agreement that his opinions and conclusions are "wrong", but not in a "dishonest" or "stupid" way as suggested here. – Thomas Klimpel Apr 22 '13 at 00:10
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    [continued] Regarding "Do modern texts on set theory bend over backwards to say precisely what is and what is not an infinite set?", don't confuse this with the axiom of infinity from ZFC. Of course, there are some sets which are explicitly blessed as finite, and other sets which are explicitly blessed as infinite. However, the "modern texts on set theory" I consulted didn't bother to discuss the problem of how to define when an arbitrary set is finite or infinite. There are texts which discuss this problem, and it turns out that different sensible definitions lead to different answers! – Thomas Klimpel Apr 22 '13 at 00:19
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    A renowned mathematician writes a mathematical paper. Top voted answer goes "hi dar i didnt read it but i now its all wong". Sorry, you do need to read it to understand the main point. Yes you did good in finding a mispelling and then dismiss the entire paper. Just like they found errors in Wiles proof. You need to be humble to understand it, to first accept that you might have misunderstood everything. The other critique is about no references, seems you guys just skipped the actual content, which is the part that matters? – Igor Ultra Apr 22 '13 at 01:42
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    @Pete: Wildberger makes demonstrably false statements about the enterprise of set theory (not even as a foundation). As most of these falsehoods are about matters that are covered in any standard graduate-level text in the area, I can only assume that he does so intentionally. My answer, if nothing else, points out exactly one of his untruths, and provides a more honest account of the actual state of affairs within set theory. – user642796 Apr 22 '13 at 04:13
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    @Thomas: I didn't compare Wildberger to Mückenheim. I stated that Wildberger names Mückenheim as a supporter. This is true, as evidenced in the third paragraph of Wildberger's page on Views and opinions mathematical.... – user642796 Apr 22 '13 at 04:16
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    @Arthur: I don't completely agree. You write: "This is a very precise formulation which one can show yields a set which is not finite." By "show" you can only mean: it follows from the axioms that an infinite set exists. – Pete L. Clark Apr 22 '13 at 05:46
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    But then what is the essential difference between your version of the Axiom of Infinity and Wildberger's? The ones I can see are (i) Your version of the axiom is a priori slightly stronger than Wildberger's, and (ii) You are waving your hands to show that the existence of an infinite set follows from your axiom, whereas it obviously follows from Wildberger's. But neither of these differences is significant, as more careful deduction from the axioms would show. – Pete L. Clark Apr 22 '13 at 05:51
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    @Pete: Wildberger's version of the Axiom of Infinity is simply: There is an infinite set. By itself, this is a meaningless, since there is no a priori definition of either finiteness or infiniteness. This is where Wildberger invokes Russell and the need to be "extremely careful about what the words ‘infinite set’ denote". So he criticises set theory for not being precise (i.e., extremely careful) based in part on a statement that can only be a deliberate mischaracterisation of the set-theoretic enterprise. [cont...] – user642796 Apr 22 '13 at 07:33
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    [...inued] Here is an outline of a completely formal approach to infinite sets (essentially stolen from the exercises at the end of Chapter 1 of Jech's Set Theory, 3rd Millennium ed): (1) define $N$ to be the smallest inductive set (or, equivalently, the intersection of all inductive subsets of a given inductive set); (2) call a set finite if it can be put into one-to-one correspondence with an element of $N$, and infinite otherwise. (3) Show that, according to the definition above, $N$ is infinite. [cont...] – user642796 Apr 22 '13 at 07:33
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    [...inued] I have now precisely defined all my terms in a manner Wildberger does not care to. We can argue about the real existence of infinite sets, but this is not what Wildberger is doing in the part I am criticising. He seems to only state that the words set theorists use are vague and meaningless, and so their work should be doubted. – user642796 Apr 22 '13 at 07:34
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    @Thomas: I think your mistrust of set-theoretic definitions of finiteness stems from various different proposals (Dedekind finiteness, Tarski finiteness, as well as various versions of Truss) known to be inequvalent unless (some version of) Choice is assumed. Make no mistake that when Asaf (who works considerably in the absence of Choice) and I (who works almost exclusively under the assumption of Choice) speak about finite sets, we are speaking about the same notion: being equipotent with an element of $\omega$. [cont...] – user642796 Apr 22 '13 at 07:48
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    [...inued] Does the existence of the non-equivalent notions of "open cover" compactness, sequential compactness, countable compactness, realcompactness, pseudocompactness, paracompactness, metacompactness, etc. make you doubt the precision with which topologists have defined compactness? Or do you agree that these different notions describe different properties, and part of the topological enterprise is to discover to which extent these different properties are related? – user642796 Apr 22 '13 at 07:49
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    @Arthur: You're right that Wildberger's Axiom of Infinity requires a separate definition of "infinite set". I think we should be a little charitable and assume that such a definition has been given elsewhere. And I agree with you that he seems to be playing "straw man" here a bit, in that he gives a nonfactually slightly sloppy take on AOI and then uses that for criticism. I don't think that's the point of his criticism though: the idea that we have essentially assumed in our axiom system that something magical occurs applies equally well to the more standard description of AOI. – Pete L. Clark Apr 22 '13 at 14:04
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    I'm glad @PeteL.Clark is saving me the trouble of pointing out that the straw-man arguments are by those reading Wildberger, much more so than Wildberger reading set theory texts. W says very clearly at several points in the essay (search string SPECIF) why he thinks the definitions of finite, infinite, set, etc are imprecise. It is ridiculous (and maybe dishonest, by the standard being applied to W.) to disregard this and say that his criticism is refuted by the presence of a formal definition in a textbook. W's argument is that those definitions are weak, not that they don't exist. – zyx Apr 22 '13 at 21:20
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    The comparisons to topology are not accurate. Number of holes (and many other numerical invariants) can be physically demonstrated, is important in very concrete situations like geometry, graph theory and complex analysis, and it was easy to give correct but ad hoc definitions in a large number of motivating cases. The topological formalization is forced by this "material experience" and the need for a precise general theory. For completed infinite sets, there was no known need or prior experience and the sets were brought into some sort of pseudo-material existence by pure assertion. – zyx Apr 22 '13 at 22:56
  • I am not a set theorist, so I might be wrong. But I think that if we replace Axiom 6 with something like "all positive integers form a set" we get an equivalent set of axioms, don't we. And while all the known paradoxes will be the same, most of the points in this paper would dissapear... For me this suggests that A6 is not the reason why there are issues with the ST.. – N. S. Apr 25 '13 at 14:59
  • @JoshuaShaneLiberman Your $x+1$ is not good, because there are more than that electrons in the universe ;) And we can always consider the distance traveled by light in a certain interval of time.... – N. S. Apr 25 '13 at 15:01
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    This is NOT a very precise definition, this is a complete joke, your argument ended with "...", so you listed 3 elements, thats 3, not infinite. How is "..." = "goes on for infinity" better then, "there exist an infinite set" ? Also, you just showed that its not finite, what happened to excluded middle? What if you just showed its not a set after all. A cow is not a finite set, so the existence of cows must prove the existence of infinite sets. – Grandmaster Jul 07 '13 at 23:42
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    This answer is too stupid, you made a bad argument worse by clouding it with "...", it should be deleted from the site. – Grandmaster Jul 08 '13 at 00:52
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    $\ldots$ is simply a convinient way of writing "continue forth in this pattern" or if there is none "it goes forth in some way we don't care about".

    We don't need to list all infinite elements because that is practically impossible, which is an irrelevant factor to infinity and mathematics. The limitations of reality has no impact on mathematics so we come up with shorthands to write what we cannot write normally while portraying the concept at hand. The arguement is correct in that wildberger is full of it.

     – Zelos Malum Oct 09 '15 at 18:50
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    @Grandmaster The definition is in the box; the "..." is only an example derivation of the first few elements to help convince the reader that the process can be continued to extract infinitely many elements from an inductive set. The definition itself is perfectly well-formed and does not need any ellipsis. In actual practice you can't extract infinitely many elements because any proof is finite, but by using this definition you can refer to infinitely large sets like $\omega$ as a completed whole without having to pull all the elements out one by one. – Mario Carneiro Jan 04 '16 at 19:16
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    @Grandmaster is correct, and Zelos's and Mario's responses are breathtakingly dogmatic. "We don't need to list all infinite elements because that is practically impossible, which is an irrelevant factor to infinity and mathematics." "In actual practice you can't extract infinitely many elements because any proof is finite...." So you "portray the concept" while admitting that you cannot in fact concretely demonstrate it, but you maintain it is a "well-defined mathematical object"? And the one who questions your belief in infinity is "full of it." Interesting.... (cont'd) – Wildcard Feb 03 '17 at 01:17
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    (cont'd) Honestly, it sounds an awful lot like a dressed-up version of, "God is everywhere and you must not question God." Only difference is you're asserting that your ellipsis ("...") proves the existence of infinite sets, rather than your Bible proving the existence of God. (Note: I have great respect for Christianity as a religion. Partly because it does not pretend to be a mathematics. You either accept it—on faith—or you don't. I have little respect for dogmatic priests, or dogmatic "mathematicians.") – Wildcard Feb 03 '17 at 01:20
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    To make the point clearer, since @MarioCarneiro points out the definition itself does not contain an ellipsis: If that were given solely as a definition of the phrase "infinite set" there would be no difficulty whatsoever about it. However, it is not simply a definition: it is a statement that there is such a set. In other words, it asserts the existence of such a set, without any proof or evidence; and the "example" that follows obscures the lack of evidence in the ellipsis. Hence W's analogy: "There is a leprechaun which can see everything." You take it on faith if at all. – Wildcard Feb 03 '17 at 01:29
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    @Wildcard Indeed, a set closed under successor and containing the empty set is the definition of an "inductive set", and it is easily shown that every inductive set is infinite, whether or not said sets exist. This is the same sort of argument as "All leprechauns are Irish" or some such. The real contentful statement is the axiom of infinity, which postulates that the domain of discourse contains an inductive set. As for whether or not these really "exist" in the ontological sense is quite beside the point; they are useful, so we help ourselves to them and clearly mark our doing so. – Mario Carneiro Feb 03 '17 at 05:50
  • @MarioCarneiro, by "useful" do you mean "interesting to contemplate," "possible to gain tenure by writing papers about," or something else? I'll grant you can have a lot of fun playing with ideas of infinite sets, and use notation to play interesting games. I do not grant it has been shown to have any definite, concrete actual meaning, nor that it is "useful" in any meaningful sense. It reminds me of the recommended "useful" topic of Aramis's thesis in The Three Musketeers: "The two hands are indispensable for priests of the inferior orders, when they bestow the benediction." (cont'd) – Wildcard Feb 03 '17 at 08:22
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    I'll reiterate from Wildberger's paper: "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." (No one denies the interest value of philosophy. But philosophy is not mathematics.) – Wildcard Feb 03 '17 at 08:25
  • By the way, you managed to gloss over (elide) an actual definition of "infinite set" while pretending to discuss them. You just said that "it is easily shown that every inductive set is infinite," but what, pray tell, does "infinite" mean if not "inductive"? "Infinite means it is not finite"—in which case, as @Grandmaster said, a cow is not a finite set; therefore it is an infinite set. – Wildcard Feb 03 '17 at 08:30
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    @Wildcard I mean that infinity can be used as a tool to reason about the finite. Of course in most, if not all, cases these results can be rewritten to avoid the infinite in the first place, but using infinity has a great simplifying effect because it has properties not shared by its finite approximations. The definition of an infinite set is simply an element of the domain of discourse which is not finite, where a finite set is a set bijective with some element contained in every inductive set (i.e. some element of the natural numbers). – Mario Carneiro Feb 05 '17 at 01:30
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    @Wildcard "All those theories, arguments and daydreams involving ‘infinite sets’ need to be recast into a precise finite framework or relegated to philosophy." Even granting all the rest of that quote, this is a normative claim that I see no reason to agree with. And "philosophy is not mathematics" is quite ironic considering that Wildberger's claim is essentially philosophical. Mathematically, there is no reason to avoid infinity if it is not inconsistent; but it is philosophically suspect by Wilberger's (ontological) arguments about "true existence" of infinities. – Mario Carneiro Feb 05 '17 at 01:34
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    @Wildcard A good video which in part discusses why infinity is useful: Hilbert's Curve, and the usefulness of infinite results in a finite world – Mario Carneiro Feb 05 '17 at 01:56
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    I am not sure I understand the specific issue discussed here. The statement "there is an infinite set" is trivially equivalent on the basis of the other standard axioms to the standard formulation of the axiom of infinity. Of course, this requires we have a definition of the word "infinite". But all the usual definitions (Dedekind infinite, not equipotent to a finite ordinal, not Tarski-finite, ...) give equivalent formulations. – Andrés E. Caicedo Sep 07 '17 at 17:55
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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.

Asaf Karagila
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    But many times we get extra interesting properties about the objects which exist in reality by studing those objects which don't exists. I wonder often how many physicist/chemists realize that when they find the best line fitting a data of 666 points (or best polynomial) they are actually doing a projection in a very artificial $\mathbb R^{666}$... Probably not even all statisticians realize the abstract algebra behind regression analysis... – N. S. Apr 09 '13 at 19:55
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    I agree, my favorite example is uses of compactness to prove claims about finite sets (e.g. Ramsey theorem, trivial for infinite sets, use compactness to deduce correctness for finite sets; constructive proofs are useless because they don't even give you tight bounds in these cases). But this is still a mind game. And if you look at how application is being done you will probably shed a tear... – Asaf Karagila Apr 09 '13 at 19:58
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    Blasphemy, $\frac{1}{2}$ exists! Call me a platonist. – Jeppe Stig Nielsen Apr 09 '13 at 22:54
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    @Jeppe: It may or may not exist as an ideal mathematical object; but it certainly does not exist in our physical reality. We can never tell with accuracy whether or not you cut the cake exactly in half. – Asaf Karagila Apr 09 '13 at 22:55
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    Nor what is one cake actually. – Francisco Presencia Apr 10 '13 at 04:58
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    @Frank: Unless you are making a Portal reference, explain yourself. – Asaf Karagila Apr 10 '13 at 06:30
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    What is 1 cake? What are 2 cakes? You cannot say either 2 cakes unless they are physically identical. Nor 1 cake, as you call it cake as an approximation to your personal ideal cake. So counting cakes is as 'real' as counting fractions of cakes, just approximations. No portal reference though... – Francisco Presencia Apr 10 '13 at 10:05
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    @Frank: So... there are no people on the planet either. No one is alive... nothing is real. Huzzah. – Asaf Karagila Apr 10 '13 at 10:10
  • @AsafKaragila, basically, although how can you say no one is alive if there's no one? And what planet? (; Okay, I'll stop now. – Francisco Presencia Apr 10 '13 at 10:15
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    Sure you can cut it in half, just count the number of particles in each piece? Count.. integers.. head explodes – Igor Ultra Apr 18 '13 at 15:57
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    @Igor: Troll. You do know that the number of particles is not static, right? Furthermore, what if everything has an odd number of particles to it? What if there are fluctuations in the size, mass, and density of protons, and other particles? – Asaf Karagila Apr 18 '13 at 16:00
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    Its static at each instant of time. But I agree that some parts of math, like set theory, is only meaningless formal games, and does not refer to real objects. And I agree that someone who only knows set theory will find it natural to extrapolate their experiences unto all of math and probably all human knowledge. – Igor Ultra Apr 18 '13 at 16:58
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    @Igor: Good thing that I know more than set theory. Is there a point to your thinly veiled insults? – Asaf Karagila Apr 18 '13 at 17:09
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    I see you've made a friend here as well! – user642796 Apr 18 '13 at 17:09
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    @Arthur: I am a great believer in making friends! Especially if they can get you pickles later in life! :-) – Asaf Karagila Apr 18 '13 at 17:11
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    I get the sneaking suspicion that IU will not be among your pickle-gatherers. ;-) – user642796 Apr 18 '13 at 17:14
  • @Arthur: I guess not. But what will happen in Oropa? How many pickles have you secured thus far? – Asaf Karagila Apr 18 '13 at 17:24
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    Asaf, to many people math is more than a mind game. Mathematics does indeed have to do with the physical world, I think. Just because abstract notions don't exist in the physical world as, say, apples do, that doesn't mean they don't exist nor have to do with it. I recommend Must we believe in Set Theory by Boolos, where he questions the existence of sufficiently large cardinals while accepting the existence of other abstract concepts. – Quinn Culver Apr 21 '13 at 14:36
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    @Quinn: Oh, of course. It is perfectly reasonable that Platonic mathematical universe exists somehow. But I think it should be fairly obvious that this is a belief, not a fact. Let alone to claim that this is the universe we are in, which amounts to religious hogwash. Beliefs are in our minds (hearts contain blood, not information!) and therefore even if we do believe in some actual existence of mathematical objects, this is still something which is a mind game -- just in a very broad meaning of "mind game". After all, I am strongly agnostic (with fatalistic nihilism on the side)... – Asaf Karagila Apr 21 '13 at 14:39
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    What do you mean by "there is no number which is 1/2"? – Quinn Culver Apr 21 '13 at 14:49
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    When you say "It doesn't have to do with the physical world.", do you mean "it has nothing to do with the physical world" or "it doesn't necessarily have anything to do with the physical world" (or something else)? – Quinn Culver Apr 21 '13 at 15:27
  • @Igor: I don't have a car. Please stop commenting on my posts if you feel that you have nothing constructive to contribute to the site. – Asaf Karagila Apr 21 '13 at 16:16
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    @Quinn: I mean that there is no accurate way to physically measure $\frac12$. As for the second comment, of course I mean it doesn't necessarily have to do with the world. Adding two apples from my left hand to one apple on the right will still amount to three apples. But the things which are in fact physically representable (provably!) are somewhat limited mathematically, at least when you accept the notion of infinity as a valid mathematical idea. – Asaf Karagila Apr 21 '13 at 16:19
  • let us continue this discussion in chat – Quinn Culver Apr 21 '13 at 17:13
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    @Quinn: I don't do chat on SE. – Asaf Karagila Apr 21 '13 at 17:21
  • Well, it seems that this conversation is not comments-appropriate, so I guess I'll just drop it. Good talk though. – Quinn Culver Apr 21 '13 at 17:47
  • @Quinn: Yes, good talk. If you ever feel like continuing it you can find me in the comments here or elsewhere. – Asaf Karagila Apr 21 '13 at 17:55
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    You should gain some humbleness towards science, and how far human knowledge has progressed, first road to knowledge is to accept that you might have misunderstood everything. – Igor Ultra Apr 22 '13 at 01:52
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    @Igor: And can you accept that, if what you believe in is "law", and it cannot be wrong? Can you accept that it might be wrong, but still believe that it cannot be wrong? That's quite a dissonance. – Asaf Karagila Apr 22 '13 at 07:00
  • @Jeppe: moreover, I know what a calf is and where I can find one, but what is a half and where can I find one? (to decode my sarcasm: but does this spoil the meaningfulness of arithmetic (numerical relations) and math in general?) – Gottfried Helms Apr 22 '13 at 12:56
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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.

Carl Mummert
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    Carl, if you look the author's page up and check his youtube videos, you may start to think this was not a tongue-in-cheek essay. – Asaf Karagila Apr 10 '13 at 08:17
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    It would take me some time to write a detailed answer, but I don't think those examples contradict what is written in the essay. They are correct only if you start from the very assumptions the essay is arguing against. For example, Whitehead problem does not show that choice of postulates is an issue in group theory. It shows that abelian group theory built on a particular set theory shares some complexities of the sets. This is not automatically a group theoretic issue, it could be the choice of set theory foundations that causes the independence, or that it is never seen in practice. – zyx Apr 20 '13 at 06:54
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    I do not think that the question whether set theory texts include a definition of "infinite set" is a matter of assumption, nor the fact that there is a computable bounded increasing sequence of rational numbers such that no computable real number can be its limit. The Whitehead problem, which arose purely as a question in group theory, is precisely an example of how independence arises in practice. – Carl Mummert Apr 21 '13 at 02:35
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    Set theory texts do not include definitions of how a finite (or infinite) set is specified, which is part of what Wildberger criticizes in the essay and includes as a part of the "defining" he says is missing. Given his other comments he seems to be aware of formulations such as the one you gave, and the possibility to write them as ZFC formulas. Those formulas and their informal equivalents do not constitute an answer to the point you quoted from the essay. – zyx Apr 21 '13 at 04:01
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    Whether the reals are complete in a constructive, computable or finitistic account (or whatever analogous account the essay might advocate) depends on the definitions of real number and of completeness that are used. I do think that in many formalizations there will be a lack of completeness for reasons such as you indicated, but there can be definitions of completeness that are classically equivalent but constructively, computably or finitistically inequivalent, where the real numbers satisfy one formulation and fail another. – zyx Apr 21 '13 at 04:09
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    Wildberger could presumably have made those arguments if he wanted to, but he did not. Instead, he claimed that set theory books don't define "infinite set" and that the computable real numbers are complete, with no further comment. This is why the essay should be treated as tongue in cheek. The plain appearance is that Wildberger went out of his way to avoid making any point clearly. Those who already understand finitism can read his essay in that light, at which point they will understand the jokes. But in evaluating the essay I look at what is literally said, not at what could have been. – Carl Mummert Apr 21 '13 at 12:27
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    But Carl, did you "think clearly about the subject for a few days"? – Quinn Culver Apr 21 '13 at 14:48
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    As hinted above, it is an interesting and controversial question whether the Whitehead problem (or any other independent statement) is an example of set theory "appearing in practice" in mathematics. Harvey Friedman, who has devoted much of his career to trying to demonstrate the relevance of set theory, has often articulated (to set theorists) what I believe to be the overwhelming consensus among mathematicians, which is that none of the famous examples are convincing. Citing Whitehead as a self-evident refutation of Wildberger's argument is restating your assumptions as a conclusion. – zyx Apr 22 '13 at 05:18
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    "Wildberger could presumably have made those arguments if he wanted to, but he did not." - a reading of the whole essay, and not only the one sentence in isolation, shows that Wildberger did make those arguments and others about lack of definition of infinite sets in textbooks. A search for the string SPECIF might help, it finds some of the passages in which he says that the standard presentations do not discuss how (or what it means) to specify a set, how to compute with sets as a data type, and other things he takes as a requirement for being well-defined. – zyx Apr 22 '13 at 05:25
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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.

Vladimir Sotirov
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    Woodin has a nice paper that makes some of these points as well: The Tower of Hano. In Truth in mathematics (Mussomeli, 1995), 329–351, Oxford Univ. Press, New York, 1998. He sometimes describes himself as a "conditional platonist". Let me quote from the paper: "By a routine Gödel sentence construction we shall produce a formula $\Omega(x_1)$ in the language of set theory which implicitly defines a property for finite sequences." "If there exists a sequence of length $n$ with this property, then $Exp_{2011}(n)$ does not exist". (Cont.) – Andrés E. Caicedo Apr 24 '13 at 23:22
  • Here, $Exp_1(n)=2^n$ and $Exp_{k+1}(n)=2^{Exp_k(n)}$ for all $k$. "However this sentence has the feature that, if arbitrarily large sets can exist, then, for each suitable $n$, there is no proof of length less than $n$ that no such sequence of length $n$ can have this property." "We shall argue that there are limitations to the extent our experience in mathematics to date refutes the existence of such sequences." (Cont.) – Andrés E. Caicedo Apr 24 '13 at 23:26
  • "In fact we shall argue that a consistent philosophical view must in effect acknowledge the possibility that the sequences of length $10^{24}$ could exist, just as those who study large cardinals must admit the possibility that the notions are not consistent." – Andrés E. Caicedo Apr 24 '13 at 23:28
14

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.

TonyK
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    Yes, he's trying to impose a sort of realism on the construction of numbers that no one ever claimed existed. To say that "$1 = { 0 }$ is ridiculous" is to beg the question of what $1$ is at all. – A.S Apr 09 '13 at 21:39
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    Although a rational number is equal to various fractions, it's pretty moronic to define it as an equivalence class. Do you define 2 as the equivalence class consisting of all expressions which evaluate to it, like 3 - 1, 4 - 2, 8/4, ... – Kaz Apr 10 '13 at 02:03
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    @Kaz: it depends on which $2$ you mean, but in $\mathbb{Z}$ constructed as the extension of the cancelative semigroup $\mathbb{N}$ to a group, $2$ is indeed the equivalence class of all pairs of naturals $(a,b)$ such that $a = b+2$. This is the standard construction of $\mathbb{Z}$ from $\mathbb{N}$. – Carl Mummert Apr 10 '13 at 02:06
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    I'm not denying that there is such a class. It's just not what comes to mind as 2. – Kaz Apr 10 '13 at 02:11
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    @Kaz Precision is a way of thinking. Defining Q from Z like that looks a little embarrassing, sure. But here are its advantages: it's precise and clear; basically the same construction generalises to localisations of arbitrary rings (it's far less obvious that these exist); understanding how R and C came from Q gave us the fields Q_p and C_p; and so on. This isn't just nonsense. Precision is a way of thinking - without these tools, some genuine mathematical problems would still be unsolved. – Billy Apr 10 '13 at 05:19
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    @Kaz: That's how I like to define $\mathbb Q$. If it makes me a moron in your eyes, I can live with that. – TonyK Apr 10 '13 at 15:56
  • Definitions are like tools, and something can have more than one definition, which is like having different tools for a different job. There is no "the" definition, but if there was, it would probably be determined by some kind of popularity vote among all of the users of numbers, not just mathematicians. – Kaz Apr 10 '13 at 18:51
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    @Kaz: You said "pretty moronic". Which was a pretty moronic thing to say, in my opinion. Are you retreating from that stance now? It's difficult to tell from your latest comment. – TonyK Apr 10 '13 at 19:18
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    I think his point about rational numbers is that collecting all equivalent ordered pairs together is an inefficient way of defining a rational number. You wouldn't actually implement it this way on a computer. But I think he's missing the point. Mathematics isn't about computational efficiency, its about conceptual efficiency. – goblin GONE Apr 12 '13 at 06:26
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    @Kaz: Don't use language like "moronic"; it's inappropriate. – Zev Chonoles Apr 25 '13 at 18:15
  • Equivalence classes are impredicative. There's good reason to reject that and use something like setoids instead. – Nathan BeDell Mar 24 '17 at 14:58
13

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!).

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    -1 On meta, you wrote things like "poor fit for this site ... to argue specifically with a well-known crankish essay, a task that is generally considered fruitless" and "Furthermore, giving serious responses to it is arguably counter-productive to begin with, by implying the original essay actually merits a direct response". I think your answer contains "debate, arguments and extended discussion", all these things why you correctly explained that such a question should be closed. Why on earth then do you write such an answer??? – Thomas Klimpel Apr 21 '13 at 15:04
  • @Thomas: As I said there, I was trying to represent the viewpoint the questioner asked about, but wasn't sure if I actually had that view (or the contrary one). At the time, it had appeared that nobody who was involved in the closing was paying attention, and so I wanted the person to get a timely answer. While I am happy to argue the case for closing this and the other question, I haven't convinced myself, and it appears the current majority opinion is that this (and the other) question should be open. –  Apr 21 '13 at 15:51
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    Do you really think MSE is a place for "debate, arguments and extended discussion"? The "majority opinion" is one thing, but the stack-exchange framework explicitly tries to minimize such things, probably for good reasons. Anyway, I was just explaining why I downvoted. – Thomas Klimpel Apr 21 '13 at 16:18
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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).

byserpas
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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."

Kasper
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    Ok, so we should not do physics. Physicists should quit their jobs. Because mathematics should not describe the real world, because it us humans who decide how the universe works. – Igor Ultra Apr 18 '13 at 16:50
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    @Igor Ultra: ??? Mathematics can be used to describe the physical world. That doesn't mean it has to be limited to such. – The_Sympathizer Apr 21 '13 at 07:40
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    Until very recently, prime numbers had never been observed in nature. Wildberger could have therefore concluded that number theory cannot be a valid branch of mathematics. – Peter Webb Mar 02 '15 at 06:57
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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 $5.$

bof
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  • Erm {{citation-needed}} for your last claim. The proof of BT involves rotations that allow reproducing hollow spheres. I don't see how to easily get larger spheres, or to double a cube for that matter. – user21820 Aug 23 '16 at 12:26
  • @user21820 For brevity let "set"="subset of $\mathbb R^3$" and "equivalent"="equivalent by finite decomposition" i.e. $A$ is equivalent to $B$ if for some $n$ the sets $A,B$ are partitioned into $n$ sets $A_i$ and $n$ sets $B_i$ with $A_i$ congruent to $B_i.$ (1) A unit ball is equivalent to the disjoint union of any finite number of unit balls. (2) If $A,B$ are balls of any size, then $A$ is equivalent to a subset of $B.$ (3) If $A$ is bounded and $B$ has nonvoid interior then $A$ is equivalent to a subset of $B.$ – bof Aug 23 '16 at 12:45
  • @user21820 (4) If two sets are bounded and have nonvoid interiors, then each is equivalent to a subset of the other. (5) If $A$ is equivalent to a subset of $B,$ and if $B$ is equivalent to a subset of $A,$ then $A$ is equivalent to $B.$ (For (5) use Banach's mapping theorem.) – bof Aug 23 '16 at 12:50
  • Hmm for (5) I don't see how Banach mapping theorem gets it, but I think the set-theoretic proof of the Cantor-Bernstein theorem works. – user21820 Aug 23 '16 at 12:57
  • @user21820 Banach's Mapping Theorem: Given maps $f:A\to B$ and $g:B\to A$ there are partitions $A=A_1\cup A_2$ and $B=B_1\cup B_2$ such that $f[A_1]=B_1$ and $g[B_2]=A_2.$ If $f$ and $g$ are injections then this proves the Cantor-Bernstein theorem. – bof Aug 23 '16 at 13:00
  • Yup that's what I got from the proof of Cantor-Bernstein theorem. I was not aware it was called Banach's mapping theorem (which Wikipedia says is about something completely different). Thanks for the clarification! – user21820 Aug 23 '16 at 13:02
  • @user21820 The terminology "Banach's mapping theorem" is used e.g. in this paper by Brualdi. – bof Aug 23 '16 at 23:04
  • Ah I see thank you very much. – user21820 Aug 24 '16 at 02:27
  • Not it's not. You would only be able to do that if the parts didn't obey some really important properties which all component parts of any reasonable solid ball must obey. For example two ENTIRELY IDENTICAL objects are the same object. The axiom of choice used in this theorem assumes to the contrary - a contradiction. – Robert Frost Dec 09 '17 at 14:48
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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.

Lehs
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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.

Doug Spoonwood
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    It might be the case that the natural numbers do not form a set, but every number still have a successor. In this case the natural numbers form a class rather than a set. – Asaf Karagila Apr 10 '13 at 08:16
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    @AsafKaragila I hadn't considered that. But is the question "do infinite sets exist?" or "do infinite collections exist?"? – Doug Spoonwood Apr 10 '13 at 11:59
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    Of course that if you are going for "nothing infinite exists" then either there are finitely many natural numbers, or you cannot really talk about the collection of natural numbers in a meaningful way. But if you consider the theory of $\sf ZF$ without the axiom of infinity, and replace it with its negation, then you have a theory which is strong enough to develop a lot of basic mathematics (it is bi-interpretable with first-order Peano), and yet there are no infinite sets there. – Asaf Karagila Apr 10 '13 at 12:01
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    The axiom of infinity exists for one reason - to make set theory interesting. Without it, there's not a whole lot to talk about. – Peter Webb Mar 02 '15 at 07:01
  • @DougSpoonwood, this answer itself is a strawman argument. Wildberger stated that the natural numbers are not finite, but questioned whether it is reasonable to assume that all natural numbers may be collected into a well-defined mathematical object. In other words, by using the phrase "set of natural numbers," you have already assumed that either the natural numbers are finite, or infinite sets are possible. – Wildcard Feb 03 '17 at 01:01
  • Let us continue this discussion in chat. – Wildcard Apr 29 '19 at 06:31