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> The approaches using equivalence classes of Cauchy sequences ... suffer from an inability to identify when two “real numbers” are the same
Perhaps there's no computable 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!
I'm not sure if he's a finitist or ultrafinitist [1] but the uncountable reals are clearly a problem for him. I'd like to argue for their usage from a "soft" point of view as opposed to just giving axioms. 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…
> Mathematics and physics both began with the advent of astronomy Arguably mathematics began with accounting and surveying and trade, and more generally with counting and measurement.
I know Norman; not personally. 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…
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...
Did anyone understand this bit? > The approaches using equivalence classes of Cauchy sequences ... suffer from an inability to identify when two “real numbers” are the same Perhaps there's no computable 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!
Earlier quoted context omitted.
> The problem with this is that it is wrong. > You can translate between different logical theories by building a model of one in another, so it's not like you loose anything. But it's cooky to insist that we should start with ZFC of all things. I don't see how these two are consistent. Almost everything most mathematicians do can be done both in ZFC and your favourite non-kooky axiom system. Certain Powers That Be s…
Slightly tongue in cheek, but the analogy to programming is this: > "Almost everything most programmers do can be done both in x86 assembly and your favorite non-kooky programming language. Certain Powers That Be seem to have decided that x86 is the foundation of computer science. [...] Why does it matter?" The problem is that it is difficult to translate results in a theory built in ZFC to other "architectures". In…
I'd make the following alternative analogy: I code in Python on an x86, because that happens to be the machine on my desk. If you told me I should be using POWER instead of x86, I'd probably just shrug: I could do that - my work is portable - but it's also completely irrelevant to my work. I think this would be how most people in say analysis, algorithms or combinatorics feel, for example.
I know Norman; not personally. 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…
After listening to Wildberger'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'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 logi…
Earlier quoted context omitted.
You're right. Classical measure theory lives in a model divorced from physical reality. You can show that all of the fancy counterexamples which necessitate the complicated constructions of measure theory are artificial (e.g., the characteristic function of a non-measurable set is uncomputable). There are better approaches to measure theory which live in different "foundations". For example, you can build measure and…
>Classical measure theory lives in a model divorced from physical reality. Everything in mathematics is divorced from reality. Unbounded integers are divorced from reality. (Once you move beyond naive realism, bounded integers are divorced from reality, but that's a deeper philosophical debate.) The only question is whether these models are more or less effective for their various theoretical and applied purposes. >…
It's a matter of degree. You can't point to a particular large integer as being "too large" to matter in real life, but there are plenty of objects in measure theory (like unmeasurable sets) that blatantly violate physical intuitions and don't seem to exist in any sense in real life.
I know Norman; not personally. 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…
> 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).