Logical difficulties in modern mathematics (2012)

Did the guy ever deliver on his promise to create a better system? I thought the problems he was pointing out was motivation for a better system (that united the naive understanding and rigor better?), but the further into the article I got, the more it just seemed like a list of complaints.<p>Having 1) studied &quot;naive statistics&quot; when I was young and 2) learning the full-on measure-theory version a la Bourbaki, and now 3) using applied statistics every day, I can say the naive version is the most useful by far. Yes, there is hand waving, but the hand waving works, and you really don&#x27;t get into that much trouble unless you try to be pathological.<p>I admit, it would be great to have a version where the naive intuition (many of which are motivated by empirics, physics, and real world situations) and the theoretical definition matched better. Even still, I don&#x27;t think the formal treatment is appropriate as an introduction -- it&#x27;s an unnecessary hazing on learning minds when they can get most of the value with a bit of hand waving.

I know Norman; not personally.<p>His real complaint as far as I&#x27;ve ever been able to determine is that he is a highly symbolic thinker; and because of that won&#x27;t accept certain assumptions that everyone else takes as a given - usually what the Reals are. I&#x27;m very happy to accept than any length in geometry is a number by definition (hence sqrt(2), constructed by a 1-1-sqrt(2) triangle, is clearly a number corresponding to that length). He won&#x27;t accept sqrt(2) as a number because it can&#x27;t be represented in Hindu-Arabic notation. This isn&#x27;t really a logical issue, he just won&#x27;t use everyone else&#x27;s definitions.<p>He&#x27;s worth listening too because he is good at maths despite that handicap and his perspective is interesting to provoke a bit of reflection on what your assumptions are and what does infinity really mean anyway. His complaints are otherwise unlikely to catch on.

After listening to Wildberger&#x27;s rants – some of which are very educational and some of which are just rants, I keep thinking that his actual problem is that he doesn&#x27;t seem to believe in the implications of the axiom of the excluded middle. He goes on about infinities and such, but the deeper issue seems to be that he thinks in a constructive, intuitionistic way whereas the majority of mathematics uses classical logic and even accept the axiom of choice.<p>I wonder why he never talks about this – surely as a mathematician he should be aware of intuitionistic logic? (Or is it that as a computer science and linguistics wannabe, I am aware of it but many mathematicians aren&#x27;t bothered to take a look?)<p>If you stop thinking of axioms as value judgements and instead as definitions of formal systems (or adopt a more general system that encompasses others, such as Gentzen style sequent calculus with the only axiom being modus ponens), you achieve a piece of mind. But obviously his beef is not only that; he would like other people to admit the value of the thinking he prefers. And I agree with that.

There is a distinction between analytically correct expositions, and ones which build on naive intuition to teach students. It is inappropriate to expect beginning students to follow a logically rigorous exposition.<p>That said, there are philosophical questions about truth, infinity, unknowable statements and the like. Mathematicians have by and large settled on a set of answers to these. Every statement is true or false, regardless of whether we know the answer, or even whether we can know the answer. Infinite sets exist, and are described by a known set of axioms called ZFC. Almost all real numbers that exist can never, even in principle, be written down or described in any meaningful way. (In what sense do they exist again?)<p>All of these statements are part of classical mathematics. Almost every elementary exposition will implicitly assume that they are try. Yet they can all be questioned, and their truth can never be settled in any absolute sense. However woe betide the student who dares question these in a math class.

I&#x27;m not sure if he&#x27;s a finitist or ultrafinitist [1] but the uncountable reals are clearly a problem for him. I&#x27;d like to argue for their usage from a &quot;soft&quot; point of view as opposed to just giving axioms.<p>Mathematics and physics both began with the advent of astronomy: Babylonians and others were curious to trace star patterns and from there both physics and math developed in tandem and influenced each other greatly. Really, neither would have developed without the other.<p>Calculus was invented for calculations in physics. This gave rise to differential equations which we use to model so many nontrivial things. The differential eqns describe flow and continuity and arguably reality. Differentiation and smoothness can&#x27;t be defined over finite sets in the same way. My philosophical counter-argument to finitists is that clearly we&#x27;re on the right path to understanding the universe and nature when using the reals. It seems foolish to shy from this because computers have trouble computing some functions. Statements like &quot;there are only finitely many atoms in the universe&quot; don&#x27;t improve our understanding of much but PDEs explain.<p>[1] <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Ultrafinitism" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Ultrafinitism</a>

If real numbers get this guy&#x27;s goat, I think he will go ballistic when he hears about bra-ket formalism that quantum theory instructors foist upon students of physics.

He is obviously a well trained mathematician so reading his pieces (of which there are many) is akin to watching a tsunami come to shore---you cannot look away yet you know that this is a tragedy unfolding.<p>This said, his complaints about set theory would have been (probably) more convincing if he stated the axioms correctly. The Infinite Set axiom does not simply say that some nebulous infinite set exists it states the existence of a set with some very specific properties. His complaints about the undefined nature of a &#x27;property&#x27; are also not above criticism: GB theory (equivalent to ZF) eliminates it completely.

When discussing a certain discipline, be it mathematics or any other, the issue of rigor and foundation are completely separate. Rigor is established if you are careful to always follow the axioms determined by the foundation. But the foundation, i.e. the <i>choice</i> of the axioms can only be established in another, lower level, discipline, which, in the case of mathematics is philosophy.<p>In the early decades of the 20th century there were furious philosophical debates about the philosophy of the foundation of mathematics (between Brouwer&#x27;s Intuitionism and Russell&#x27;s Logicism, and also Hilbert&#x27;s compromise of Formalism), but they all stemmed from an underlying assumption that mathematics gains its validity from some notion of philosophical truth. One could argue to no end on how the truth of mathematics is established, but a different perspective later emerged that avoids this debate altogether: that mathematics takes its validity not from truth but from utility (which is always relative to a specific task). We cannot say that one of those views is unacceptable, and if your position is that mathematical validity stems from utility, we cannot tell you that your foundation is shaky if its utility is established.

Did anyone understand this bit?<p>&gt; The approaches using equivalence classes of Cauchy sequences ... suffer from an inability to identify when two “real numbers” are the same<p>Perhaps there&#x27;s no <i>computable</i> general method, but it seems like it highlights an even bigger problem which remained unspoken. There is an inability to tell when a sequence of numbers approaches zero!

A large part of his article seems concerned with the deficiencies and the lack of rigor in first year calculus textbooks. But I think, pedagogically, isn&#x27;t it appropriate to introduce a subject intuitively and hand-wavingly, before proceeding to a more rigorous treatment? There&#x27;s a reason students learn calculus before analysis.

There are MANY foundations for set theory, some of them very different from each other. It&#x27;s completely pointless to treat the subject as briefly as this post does.

I&#x27;m curious to hear what this guy thinks of the esoteric math in higher level economics... how base assumptions there are built and extrapolated on to ultimately guide economic policy. He would have a field day with the assumptions used in DSGE (which there are empirics to contradict those assumptions, and economists keep using those assumptions anyway).

Tried watching his video on limit ... but what was said prevented me from wasting my time further. Contrary to the claim of math foundation, it is apparent that the guy does not understand math despite being a stats professor. It is not even wrong.

I would start with identity, that&#x27;s all everything boils down to, and that math is as much a language, a shorthand to express ideas and communicate across the world like some state department, as it is computational.<p>Then I would explain that imaginary numbers mean harmonics. exponential functions give you a derivative, one function can serve as input to another, PID control, ... ... and that&#x27;s about it.