[numpy-thagoras]

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 about infinities existing are logically unnecessary to do all the heavy lifting of the math that we are familiar with. Wildberger's point (and the point of all ultrafinitist claims) is that it's an intellectual and pedagogical disservice to teach and speak of, e.g. Real Numbers, as if they're actually involving infinite quantities that you can never fully specify. We are always going to have to confront the numerical methods part, so it's better to make teaching about numbers methodologically aligned with how we actually measure and use them.

I have personally been working on building various finite equivalents to familiar math. I recommend anyone to read Radically Elementary Probability Theory by Nelson to get a better sense of how to do finite math, at least at the theoretical level. Once again, on a practical level to do with directly computing quantities, we've only ever done finite math.

[ants_everywhere]

I like to imagine at the end of the human race when the sun explodes or whatever, some angelic being will tally up all the numbers ever used by humans and confirm that there are only finitely many of them. Then they'll chalk a tally on the scoreboard in favor of the ultrafinitists.

As long as someone isn't a crank (e.g. they aren't creating false proofs) I enjoy the occasional outsider.

[numpy-thagoras]

Heh that's about the only place and time when we'll know for sure, and until then, it's just high-grade banter :)

[griffzhowl]

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 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.

[alexey-salmin]

> 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.

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.

[griffzhowl]

I was thinking more about applications in physics where calculus and irrational quantities are used all the time.

At more advanced levels the theories are based on differential geometry and operators on Hilbert space. I'm not sure if fully worked out finitist versions of these even exist. Where finitist versions do exist, they're often technically more difficult to use than the standard versions, which is the opposite of an improvement in my view.

Whether it's undesirable for your mathematical foundation to prove the Banach-Tarski paradox is debatable. It's counter-intuitive, but doesn't lead to contradictions, as far as is known. It doesn't apply to physics because the construction uses non-measurable sets.

[alexey-salmin]

I'm not a finitist myself but my understanding is that it has to do as much with physics as does ZFC, which is very little. The math used in physics works on practice and did work long before the question of foundations even came up.

The problem that bothers some mathematicians is that despite working well math still lacks a solid foundation. Furthermore it's basically proven that these foundations can't even exist, or at least for the mainstream version of math. This is where non-mainstream versions pop up. The denial of uncountable sets does help you resolve some of the paradoxes. Not all unfortunately, even the countable sets already lead to things like incompleteness theorems. Well, one can dream.

[griffzhowl]

> Furthermore it's basically proven that these foundations can't even exist,

What are you referring to? The current working foundation is ZFC but there are equivalent type theoretical foundations like what Lean and other proof-checking software uses. I guess you know that, but that's why I don't know what you mean by saying this

[alexey-salmin]

I'm referring to the failure of Hilbert's program. All the incompleteness, undefinability and undecidability results arise when and only when some sort of infinite objects are present so I can definitely see the allure of finitism.

ZFC is a working foundation of math but it's unknown whether it's consistent or arithmetically sound and important statements like CH are independent from it. It's a "working foundation" but not a "true foundation" which alas cannot exist.

As mentioned above I'm personally not a finitist and think that math without infinite and uncountable sets is intellectually poorer. I don't mind however developing further a finitist subset of math and see what's provable (and describable) in it, much like there's value in proving theorems in ZF instead of ZFC whenever possible.

[fuzzfactor]

>An eventual output of a calculation has to be a finite result, but the concepts that we use to get there are often not.

This is so true but it can be good if you're flexible enough to try it either way.

With massive tables of physical properties officially produced by pages of 32-bit Fortran it really did look like floating-point was ideal at first. Because it worked great.

The algorithm had been stored as a direct mathematical equation, plain as day, exactly as deduced with constants and operations in 32-bit double-precision floating point.

But when the only user-owned computers were still just 8-bit machines, there was no way to reproduce the exact results across the entire table to the same number of significant figures, using floating point.

Since it's a table it is of course not infinite, and a matrix to boot. A matrix of real numbers across an entire working spectrum.

The algorithm takes a set of input values, calculates results as defined, and rounds it off repeatably in the subsequent logic before output, so everyone can get agreement. The software OTOH takes a range of input values and outputs a matrix. And/or retains a matrix in "imaginary" spreadsheet form for later use :)

Every single value in the matrix is a floating-point representation of a real number, but they are rounded off as precisely as possible to the "exact" degree of usefulness, making them functionally all finite values in the end. This took a lot of work from top mathematicians, computer scientists, and engineers. And as designed, the matrix then carries the algorithm on its own without reference to the fundamental equation.

The solution turned out to involve working backward from the matrix reiteratively until an alternate algorithm was found using only integers for values and operations, up until the final rounding and fixed.point representation at the end. Dramatically unrecognizable algorithm but it worked and only took 0.5 kilobytes of 8-bit Basic code which was a fraction of the original Fortran.

This time the feature that showed up without having to make extra effort was the property of being more precise based directly on increased bitness of the computer, without need for floating-point at all. Of course the Fortran code accomplished this too by the wise use of floating-point but it took a lot bigger iron to do so. And wasn't going to be battery powered any time soon way back then.

>somehwere this finitist foundation disagrees with the results obtained by the standard foundation,

>there's no reason to think the standard foundation is in error.

This is "exactly" how it was. There were disagreements all over the place but they were in further decimal places not representable by the table. The standard was an international standard having carefully agreed-upon accuracy & precision, as defined by the Fortran which really worked and was then written in stone, with any nonmatched output being a notable failure.

[dr_dshiv]

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?

[alexey-salmin]

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 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.

[jsbg]

> 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

[alexey-salmin]

That is my point, yes.

[drdec]

If you do not accept that space is infinitely divisible, then the diagonal of a unit square does not actually exist in the space.

[adrian_b]

A long time has passed since the paradoxes of Zeno of Elea, so now there really is no reason for not accepting that space is infinitely divisible.

The error of Zeno of Elea was that he did not understand the symmetry between zero and infinity (or he pretended to not understand it).

Because of this error, Zeno considered that infinity is stronger than zero, so he believed or pretended to believe that zero times infinity is infinity, instead of recognizing that zero times infinity can be any number and also zero or infinity.

For now, there exists no evidence whatsoever that the physical space and time are not infinitely divisible.

Even if in the future it would be discovered that space and time have a discrete structure, the mathematical model of an infinitely divisible space and time would remain useful as an approximation, because it certainly is simpler than whatever mathematical model would be needed for a discrete space and time.

[drdec]

> For now, there exists no evidence whatsoever that the physical space and time are not infinitely divisible.

What is your evidence for it? You want to make a claim about something being infinite, it is up to you to provide evidence.

> recognizing that zero times infinity can be any number and also zero or infinity.

This statement makes no sense in formal mathematics. Multiplication is a function, which means for each set of inputs there is one output. I imagine you are trying to say something about limits here, but the language you are using is very imprecise.

[numpy-thagoras]

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.

[birn559]

> 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.

[numpy-thagoras]

It's part of a dependency relation, the function computes and produces an output that we call sqrt(2).

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."

[adrian_b]

I wonder what ultrafinitists do about topology.

Topology, i.e. the analysis of connectivity, is built upon the notion of continuity and infinite divisibility, which seems to be difficult to handle in an ultrafinitist way.

Topology is an exceedingly important branch of mathematics, not only theoretically (I consider some of the results of topology as very beautiful) but also practically, as a great part of the engineering design work is for solving problems where only the topology matters, not the geometry, e.g. in electronic schematics design work.

So I would consider any framework for mathematics that does not handle well topology as incomplete and unusable.

Ultrafinitist theories may be interesting to study as an alternative, but the reality is that infinitesimal calculus in its modern rigorous form does not need any alternatives, because it works well enough and until now I have not seen alternatives that are simpler, but only alternatives that are more complicated, without benefits sufficient to justify that.

I also wonder what ultrafinitists do about projective geometry and inversive geometry.

I consider projective geometry as one of the most beautiful parts of mathematics. When I encountered it for the first time when very young, it was quite a revelation, due to the unification that it allows for various concepts that are distinct in classic geometry. The projective geometry is based on completing the affine spaces with various kinds of subspaces located at an "infinite" distance.

Without handling infinities, and without visualizing how various curves located at infinity look like (as parts of surfaces that can be seen at finite distances), projective geometry would become very hard to understand, even if one would duplicate its algorithms while avoiding the names related to "infinity".

Similarly for inversive geometry, where the affine spaces are completed with points located at "inifinity".

Such geometries are beautiful and very useful, so I would not consider as usable a variant of mathematics where they are not included.

[kevin_thibedeau]

> numbers methodologically aligned with how we actually measure and use them.

We use numbers in compact decimal approximations for convenience. Repeated rational series are cumbersome without an electronic computer and useless for everyday life.

[numpy-thagoras]

The point is not about restricting what notational conveniences you prefer.

The point is to not confuse the notational convenience with the underlying concept that makes such numbers comprehensible in the first place.