His real complaint as far as I've ever been able to determine is that he is a highly symbolic thinker; and because of that won't accept certain assumptions that everyone else takes as a given - usually what the Reals are. I'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't accept sqrt(2) as a number because it can't be represented in Hindu-Arabic notation. This isn't really a logical issue, he just won't use everyone else's definitions.
He'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.
>He won't accept sqrt(2) as a number because it can't be represented in Hindu-Arabic notation.
You sure of that? Based on another of his other blog posts [1], his objection seems to be about uncomputable real numbers. Very roughly, a real number R is computable iff there exists a Turing machine that, given a natural number n on its initial tape, terminates with the nth digit of R. See [2] for a formal definition. Sqrt(2) and all familiar real numbers are computable. Of course, since there is only a countable infinity of Turing machines, but an uncountable infinity of reals, some reals must be uncomputable. Some versions of constructivist mathematics do differ from standard mathematics by rejecting the uncomputable reals and instead defining "real numbers" in such a way that they are essentially the computable reals.
Yeah, pretty confident. He doesn't like the computable or uncomputable reals; although I suspect he would like the uncomputables less.
Eg, "These phoney real numbers that most of my colleagues pretend to deal with on a daily basis ... such as sqrt(2), and pi, and Euler’s number e." [0]
Even in the article you cite, the irony of a pure mathematician of all people complaining that a concept has no tangible link to reality is a bit of a give away that he is speaking from the heart rather than the head. That isn't a valid complaint about pure mathematics; the point is patterns for patterns sake. So what if there are no known examples of your pattern? Study it anyway!
Great case of the flaw maketh the masterpiece; apart from that one little quirk with infinite things he is a lovely character and a force to be reckoned with. And I expect his personality motivates a lot of interesting research from him regardless.
I watched a couple of videos of his "Foundations of Maths A" series and wasn't very impressed (even with my limited undergraduate knowledge). For instance his arguments in his videos about set theoretic constructions aren't very rigorous or convincing. It's like he missed all the developments in category theory, type theory and logic w.r.t. those topics.
Then I watched a couple of his more advanced videos and (from my limited watching) saw that he seems not so crazy after all. It's just that he seems to like natural numbers and finite constructions a lot, although I didn't really fact check that much. Infinite and more abstract structures _abound_ (it's in their nature :P) in mathematics obviously and can be encoded symbolically just fine.
Seems fine by me, finite structures are very important as well and you can make reasoning about them very rigorous. It's just, maybe he shouldn't be teaching about all those other kinds of topics...
> I'm very happy to accept than any length in geometry is a number by definition
Interestingly, that set of numbers is still very incomplete relative to what we expect to be able to talk about in modern math. It doesn't even include the roots of all polynomials (for example, the unique positive solution to x^3 - 2 = 0 isn't the length of any constructible segment in classical geometry).
What does 'in geometry' mean in this context? There are numbers which are not constructible with compass and straight-edge constructible but which are constructible by other means (e.g. 2^(1/3) is origami constructible).
Sqrt(2) is algebraic, and can therefore be constructed fairly explocitly. Starting with integers, you can construct the rationals as an equivelence class of ordered pairs of integers with a particular definition of addition and multiplication. From their you can define polynomials with rational coeficients, and from there you can define quotient fields, Q[x]/<x^2-2> which contains two elements whose square is 2, and is isomorphic to a subset of R (eg, behaves as you would expect numbers to). The most problamatic step of the above is defining polynomials.
Can't you just define polynomials as finite sequences of rationals with nonzero last term? E.g. encode x^2 + 2x -1/2 as (-1/2, 2, 1), and define evaluation at a given point in the obvious way.
You generally define them as infinite sequences with a finite number of non-zero terms. Since we only care about a specific quotient field you could force it to work by working with only polynomials of bounded degree (<=2 should work. <=4 is probably the least bad option), but it would get ugly.
You could also ignore polynomials entirely. Since Q(sqrt(2)) is just a two dimensional vector space over Q, we could define it as the ordered pairs Q^2 with an appropriate definition of multiplication and division (like we often define the complex numbers to highshoolers), but this also gets ugly.
I guess the moral of this post is 99% of the time a finitist or constroctivist complains you can rework your theory into a more ugly one that avoids the complaint.
I am curious as to what his view would be on, say, the positive solution to the 3n+1 problem (i.e. every 3n+1/2 sequence eventually becomes a cycle). Some of these sequences may be profoundly long, so long that in his view, they may not exist. Or what if someone proves that 3 appears only finitely many times in the decimal expansion of sqrt{2} (God forbid!). How is the proof of something simple, say the solution of the Koenigsberg bridge problem by Euler any different: after all he did not enumerate all the possible paths! Even if he could, it is easy to find a manageably sized graph with too many possible paths.
His real complaint as far as I've ever been able to determine is that he is a highly symbolic thinker; and because of that won't accept certain assumptions that everyone else takes as a given - usually what the Reals are. I'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't accept sqrt(2) as a number because it can't be represented in Hindu-Arabic notation. This isn't really a logical issue, he just won't use everyone else's definitions.
He'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.