41–50 of 66 posts
Earlier quoted context omitted.
He doesn't work with imaginary numbers, either. He treats complex numbers as matrices of rationals.
Which is the same thing for all intents and purposes. An ultrafinitist is still allowed to call that 'i'.
Alright, I'll bite: To defend Wildberger a bit (because I am an ultrafinitist) I'd like to state first that Wildberger has poor personal PR ability. Now, as programmers here, you are all natural ultrafinitists as you work with finite quantities (computer systems) and use numerical methods to accurately approximate real numbers. An ultrafinitist says that that's really all there is to it. The extra axiomatic fluff abo…
So what is the length of the diagonal of a unit square, if not square root of 2? It can’t be rational—how is that rationalized by Wildberger?
We think we study the real numbers but it seems we can't even have a system to express them. And indeed, that's not even a limitation of algebraic systems: any notation over a finite alphabet can only express a countable set of distinct objects which amounts to nothing when real numbers are concerned.
I'm not a finitist, but I do find it curious that we approach mathematics by inventing a more-than-infinite set of objects that's impossible to fully grasp. I don't see it as a bad thing though, I also love Complex Analysis and many people (and some mathematicians even) denounce them for being imaginary. My impression is that transcendental numbers are as imaginary as are imaginary numbers, it's just we don't notice. And they're obviously still useful as are the complex numbers.
Earlier quoted context omitted.
Right, but to be clear, it's not that ultrafinitists like Wildberger believe that they can express all the real numbers; rather, they believe that those inexpressible real numbers don't actually exist.
How does that work for calculus which regularly looks at the limits of functions as x approaches infinity and has very real real world applications that stem from such algorithms?
E.g. when you calculate the area of a plot of land do you take into account the curvature of the Earth? You have to make a bunch of compromises in the first place to even talk about what the area of a plot land means.
Math is a bunch of useful systems that we humans have devised. We tend to gravitate towards the ones that help us describe and predict things in the real world.
But there is plenty of math which doesn't do either. It's just as real as the math that does.
Alright, I'll bite: To defend Wildberger a bit (because I am an ultrafinitist) I'd like to state first that Wildberger has poor personal PR ability. Now, as programmers here, you are all natural ultrafinitists as you work with finite quantities (computer systems) and use numerical methods to accurately approximate real numbers. An ultrafinitist says that that's really all there is to it. The extra axiomatic fluff abo…
The standard way of setting up calculus involves continous magnitudes, hence irrational quantities, and obviously that's used all over physics and there doesn't seem to be a problem with it.
I think to make a compelling case for a finitist foundation for maths you would at the least have to construct all of the physically useful maths on a finitist basis.
Even if you did that, you should show somehwere this finitist foundation disagrees with the results obtained by the standard foundation, otherwise there's no reason to think the standard foundation is in error.
Alright, I'll bite: To defend Wildberger a bit (because I am an ultrafinitist) I'd like to state first that Wildberger has poor personal PR ability. Now, as programmers here, you are all natural ultrafinitists as you work with finite quantities (computer systems) and use numerical methods to accurately approximate real numbers. An ultrafinitist says that that's really all there is to it. The extra axiomatic fluff abo…
So what is the length of the diagonal of a unit square, if not square root of 2? It can’t be rational—how is that rationalized by Wildberger?
Earlier quoted context omitted.
Read Wildberger if you want to know what he thinks. I can tell you that it is the output of a function, not a distinct entity that exists on its own independently of the computation. The whole point is that as a theory for the foundations of mathematics, you do not need to assume numbers with infinitely long decimal expansions in order to do math.
> I can tell you that it is the output of a function, not a distinct entity that exists on its own independently of the computation. Could you elaborate? What is the output of that function if not an entity in it's own? Having studied math with philosophiy minor long time ago I am curious.
On the other hand, using the axioms of ZFC, one can say any real number exists without having a function to compute it, or a proof to construct it.
For an ultrafinitist, or any finitist for that matter, we say that you only need the minimum of ingredients to produce math -- you do not need to assume anything over and above that, as it's not even helpful in the verification process.
So assuming only finitely many symbols and finitely many numbers, I can produce what we call sqrt(2). We only ever verify it numerically and finitely anyways. We can never reach decimals at infinite ordinals.
So it makes no sense to say, "Hey I assume transfinitely many entities, and my assumption says these numbers exist even though the proofs and decimal expansions are only ever finite."
It's probably worth adding the context that Wildberger's agenda is to ground mathematics in integers and rational numbers, eliminating those pesky irrationals Euclid introduced, because reasoning about them invariably involves infinities or universal quantifiers, which everyone agrees are tricky and error-prone, even if they don't agree with Wildberger's radical variety of finitism. So he was delighted to find a kind…
Thank you for this expansion. I was about to rabbit hole on how it could be that ratio-based trig (and what is that?) is more accurate than modern calculations. Re: rationals, I mean there's an infinite number of rationals available arbitrarily near any other rational, that has to mean they are good enough for all practical purposes, right?
For practical purposes, they’re bad. Denominators tend to explode when you do a few operations (for example 11/123 + 3/17 = 556/2091), and it’s not easy to spot whether you can simplify results. 12/123 + 3/17 = 191/697, for example.
You can counteract things by ‘rounding’ to fractions with denominators below a given limit (say 1000) but then, you likely are better of with reckoning with a fixed denominator that you then do not have to store with each number, allowing you to increase the maximal denominator.
For example (https://en.wikipedia.org/wiki/Farey_sequence), there are 965 rational fractions in [0,1] with denominator at most 10 (https://oeis.org/A005728/list), so storing one requires just under 10 bits. If you use the fractions n/964 for 0 ≤ n ≤ 964 as your representable numbers, arithmetic becomes easier.
Earlier quoted context omitted.
So what is the length of the diagonal of a unit square, if not square root of 2? It can’t be rational—how is that rationalized by Wildberger?
I don't know about Wildberger specifically, but an interesting point is that only a countable subset of real numbers can be described like that. Polynomials with integers coefficients are countable and so are their roots, which means almost all real numbers are transcendental. We think we study the real numbers but it seems we can't even have a system to express them. And indeed, that's not even a limitation of algeb…
Definable numbers like 2, pi, or Chaitin's constant [0] are countable. The reals are only uncountable because of numbers we can't even talk about.
Earlier quoted context omitted.
I don't know about Wildberger specifically, but an interesting point is that only a countable subset of real numbers can be described like that. Polynomials with integers coefficients are countable and so are their roots, which means almost all real numbers are transcendental. We think we study the real numbers but it seems we can't even have a system to express them. And indeed, that's not even a limitation of algeb…
> which means almost all real numbers are transcendental Definable numbers like 2, pi, or Chaitin's constant [0] are countable. The reals are only uncountable because of numbers we can't even talk about. [0] https://en.wikipedia.org/wiki/Chaitin%27s_constant
Alright, I'll bite: To defend Wildberger a bit (because I am an ultrafinitist) I'd like to state first that Wildberger has poor personal PR ability. Now, as programmers here, you are all natural ultrafinitists as you work with finite quantities (computer systems) and use numerical methods to accurately approximate real numbers. An ultrafinitist says that that's really all there is to it. The extra axiomatic fluff abo…
An eventual output of a calculation has to be a finite result, but the concepts that we use to get there are often not. The standard way of setting up calculus involves continous magnitudes, hence irrational quantities, and obviously that's used all over physics and there doesn't seem to be a problem with it. I think to make a compelling case for a finitist foundation for maths you would at the least have to construc…
Well these are probably easy to find even now? E.g the Banach-Tarsky paradox is unlikely to be provable in finitist math which is somewhat of an improvement.