A nuclear bomb just went off in the world of mathematics, and no one seemed to take note

>I am a rationalist philosopher and author working outside of academia.

not science or math

sent the guy an email,

will report back if he does it soon

>steve-patterson.com/cantor-wrong-no-infinite-sets/
Just wanted to add one of my pet peeves.

> When I write, “Jane ran to the store”, I am communicating a concept to the reader. When I write, “2x + 2x = 4x”, I am doing the same. I am saying, “There is a logical relationship between ‘having two of something’, ‘adding two of the same thing to it’, and ‘ending up with four of that thing’.”

That's just wrong. There is no "logical necessity" that putting 2 (physical) objects next to 2 (physical) objects results in 4 physical objects. That is not a mathematical truth. That's an empirical truth. An example will suffice:

Imagine you are in outer space. Drop a rock outside the spaceship, so the rock is moving at the same velocity as the spaceship. Then accelerate the spaceship for a brief moment, so that you are now at a relative 10 m/s compared to the first rock. Then, drop a second rock, so it's at rest relative to your current velocity. Then, do another brief acceleration so that you're moving at 10 m/s compared to the second rock. What is your velocity relative to the first rock? Wildberger seems to be arguing that it's a logical truth, a mathematical truth, that the answer is 20 m/s, but it's not. In the real world, the actual answer is something very slightly less than 20 m/s. However, even with that empirical knowledge, 10 + 10 still equals 20.

Math really is just playing games with symbols on paper. The application of math to the real world happens in the form of empirical models. We never show that the math of the model is right or wrong by appealing to evidence. The truth of the math is independent of any particular observation. However, by appealing to observation, we can show that the math is not an /applicable abstract model/ to this element of the real world.

>Mind of God
>Infinite
>Implying God's machine mind can conceive of anything other their beautiful, quantized, finity
>Implying we can truly know the mind of the creator
>Implying we aren't made in the creator's image

Thanks. Please be sure to see this post of mine:
where I lay out my general position, and what is wrong with Wildberger's position.

>Platonism

Actually, as I read his "paper" more, it seems he's not a Platonicist. Or at least he's using very clear language against that position. I'm surprised.

Wildberger is so fascinating. I'm continuing to read.

But yeah - I think that's the problem. He keeps bringing up metaphysics, which is just silly. He's not a platonicist, but he does have some particular metaphysical baggage. He refuses the position that math is just playing games with symbols on paper, but that's what math is.

>He refuses the position that math is just playing games with symbols on paper, but that's what math is.
As surprised as I am to see myself saying it, I completely disagree.

The universe is mechanical. There are not infinite possibilities for interactions or functional states of a given system. These laws form a sense of embedded "logic" of the universe. No attribute of a machine cannot be defined within finite bounds. At such a point, the universe is simply a finite state machine the scale of which is defined arbitrarily.

Put more directly, the universe only affords a limited number of novel actions. The machinery of the mind as well, is finite. But this is irrelevant. The universe's laws only allows for a finite spectrum of logic, regardless of the machine generating it. You can only "play games with symbols on paper" to a certain extent, there are only so many games to be played and so many ways to play the playing.

I didn't want to imply in the mail whether or not he's a platonist

And I'm sorry, but you lost me.

Let me put it like this. The entire debate in my mind comes down to this: If you accept the axioms of ZF, and if you accept that math can be and is rigorous by applying the standard rules of logical deduction on top of the axioms of ZF, then there is no problem.

Wildberger's entire problem with my position is that he doesn't like one of the axioms of ZF. He doesn't like the axiom of infinity. He doesn't like it for metaphysical reasons. Wildberger is in complete agreement with me that after accepting - what he would call an absurdity - the axiom of infinity, then the rest of modern mathematics follows in a clean, nice, and objective way.

Wildberger doesn't like the axiom of infinity, because of his particular metaphysical preconceptions. I'm totes ok with it, because the rules of the game are well specified. And further, I really like it because it's easy to work with, and because it's a really useful tool for creating empirical abstract models of the real world.

>And I'm sorry, but you lost me.
Feels subpar, man. But subpar is somehow par for the course.

I'll just go do other things instead of explaining further. Continue on with the thread.

Ok...
Not sure if I've just been dissed or not...

are you sure? The universe itself could be infinite

>it's a really useful tool for creating empirical abstract models of the real world.
Any mathematics that usefully models the real world must be computable, and can therefore be done without invoking infinity.

See, this is something that Wildberger says which is at best confusing to the reader, and at worst outright wrong.

In order for something to be computable, there must be a Turing machine that can execute and produce that result in a finite number of internal execution steps.

With the axiom of infinity of ZF, the axiom itself is formulated with a finite number of symbols. The usual proofs of Basic Real Analysis are themselves done with a finite number of symbols, with a finite number of derivation steps, that can be performed by a real and finite Turing machine in a finite number of steps.

One does not need to construct an infinite set in order to do proofs about infinite sets. That's what the axiom of infinity allows. Constructing an infinite set without the axiom of infinity is logically impossible. We simply assume that there is an infinite set, and then do work on it, such as Real Analysis.

>In order for something to be computable, there must be a Turing machine that can execute and produce that result in a finite number of internal execution steps.
pls stop posting none of what you posted is correct and this is the line.

There is nothing infinite in nature. Math is just fiction.

>The universe is mechanical
See I agree with this...and I don't see how you can better describe the Universe, EXCEPT with the infinite set axiom.
You throw out all analysis, most of calculus and differential equations. You can't really talk about variational principles, which are usually more intuitive than their differential forms.

I love rational trig btw. I just don't see it filling in for modern analysis techniques that are essential for describing physics

Uh, what? What's your problem?

In standard computability theory, in order for a function to be effectively calculable, e.g. computable, the function description itself must be finite, and for all inputs X, the function must produce an answer X within a certain finite number of steps of execution. In other words, the execution time must also be finite.

Maybe you're confusing the formal concepts of "computable" and "computable number". They're different things.

en.wikipedia.org/wiki/Computable_number

For further reading, also see:
en.wikipedia.org/wiki/Computable_function

>"There must be exact instructions (i.e. a program), finite in length, for the procedure."

>"If the procedure is given a k-tuple x in the domain of f, then after a finite number of discrete steps the procedure must terminate and produce f(x)."

We want more from our models of the physical world than merely being able to prove certain statements about it. We want to, given (possibly statistical) information about an initial state, to predict at least the observable part of the final state, or at least its probability. So we create models where the infinite can be approximated by something finite. If it was no longer possible to have a "finite stuff only" version of the model, it would be a serious limitation to what science can know. I'm not saying it's impossible we won't run into such a barrier, but I hope we don't, and I hope that if we ever end up stuck on such a model, we'll keep looking for a more complete picture of reality.

Let 'a' be the largest number.
Let b=a+1.
Since
b>a, a is not the largest number and by contradiction there is no largest number.

Google it yourself and you can find a better one.

Sure.

Looks like he doesn't understand what infinity means...

Once you declare your axioms, there's nothing fundamentally new. The people who then try to argue against it are simply declared cranks and that's the end of it; even though they usually are they are still denied any benefit of the doubt. Mathematics is just boring and mostly closed minded in this sense. There are mathematicians who, if they took the shit they spewed seriously, would not leave a burning building because the definition of "fire" was different.

It could be. Doesn't really matter if it is though. A machine beyond a given scale likely cannot produce novel logic, it can only duplicate capacity for logic it already has. If it can infinitely produce logic that roughly increases with its scale, it would be too slow and flimsy to be useful or at all lasting. Therefore the universe indirectly disallows it.

I think it's best framed as a spectrum of affordances. The universe, via its laws, is not capable of allowing infinite affordances. At least on our scale.

You weren't. At least not intentionally.

No, it couldn't.
It could be unbounded, but certainly not infinite.

>they are derived from the laws of logic, and they apply to all universes that are composed of existent things.

This is what happens when non-logicians try to into logic.

Doesn't "infinity" imply beyond human comprehension? Just because we get bored counting/die doesn't mean there aren't theoretical digits beyond that last point.
That's like saying the universe just abruptly stops precisely where our augmented vision halts.

The physical universe may be finite, but that doesn't mean we can't act mathematically beyond that limitation.
Let's say M = all the measurable matter and energy in the universe.
What is M*2?

>Doesn't "infinity" imply beyond human comprehension?
Its not that hard to comprehend

Yes exactly, math is entirely theory. Numbers are separate from whatever physical properties they describe. Our universe may be finite as said in but the mathematics don't.

I don't get what you expect him to write back with. You basically talked about yourself a lot and then said you reject his argument because he's not offering an alternative.

Platonism is correct.
This guy isn't.

Thank you user. I knew it would be something retarded (or it actually would've been news).

It's rather troubling that someone struggles so hard to understand the idea of abstractions.

This is hilarious

Its not playing games with symbols on paper but once you establish axioms and rules which are derived from them the symbollic games are the same as the logical\mathematical steps. This is why symbols and notation exist.they're like function libraries in programming you keep so you dont have to redo everything all the time.

>Implying that science even takes the existence of a god into consideration.
You're right tho.

Does he believe in real numbers though?

I fucking hate constructivists and finitists, how can one study maths for so long and not realize that we use it to model and describe the world exactly because it is not bound to the physical world. Get your fucking shit together, you contributed nothing to maths by asking questions that were witty in high school.

I don't believe 1, 2 and 3 are metaphysically real. So what?

As far as I can see I didn't reject any argument, unless you could "what they are doing is not math" as argument that "infinite sets don't exist". I don't think infinite sets "exist" either. I also certainly didn't reject anythung of a lack of alternative proposals.
The point of the mail is debatable. I mostly want to see if he agrees that he rejects formalism in any case, in which case the case is clear.

You can't know if the universe follows any laws precisely (in the sense that there is a physical theory that descibes the ujiverse at all scales)

The guy of OPs rant doesn't throw away the infinite, just infine sets. He stills wants to keep a "without bounds" notion, and btw. there is constructive analysis, for exam Pyl

Uhmm, sir, what the fuck is you doing?

Axioms are not up for discussion

If you want, form your own theory with your own axioms, but please GTFO of my internets

t. genius IQ 178

Proof that infinity exists:

You have a certain amount of energy
but you could have half that amount
and half that amount
and half that amount
you can do this an infinite number of times

>The guy of OPs rant doesn't throw away the infinite

but he does

Here's where he does it:

>Thus, it’s clear: the modern world desperately needs new foundations for mathematical reasoning. Math needs to be logical – grounded in the principles of identity, non-contradiction, and clear conceptual reasoning – and it also needs to be metaphysically precise. We need to eject infinities, Platonism, and Cantorism from all of mathematics, and relegate them to the world of mysticism and Numerology.

>WE NEED TO EJECT INFINITIES

>Human mind can not comprehend scale of infinity
>it doesn't exist
what.

I have a ground-breaking idea.

We call "Pure Mathematics" "Theology" and we call "Applied Mathematics", "Mathematics".

I mean, most people do that already, but I figured it should be public knowledge by now, instead of keeping it a polite secret.

"Pure maths" should be called 'Philosophy' rather than 'Theology'.

So when anyone finds an application for some piece of pure mathematics, they're basically stealing fire from gods? I like the idea.

>implying an axiom is "wrong"

It's sad because that guy seems pretty intelligent but somehow does not seem to understand that words in math are not the same as words in English, even though they are spelled the same

>Another point about language. It’s dangerous to uncritically use terms like “divide” when talking about mental conceptions. That’s a term borrowed from the physical world, where objects can be divided into their constituent parts – like a big ball of clay being split into two small balls of clay.
>Numbers do not work like this. They are not composite objects. “4” is not actually composed of “two 2’s”, or “four 1’s”. “4” is a symbol for a discreet concept that we can actively manipulate in our minds.
>While we can practically “divide 4 by 2”, metaphysically speaking, that “division” didn’t happen until you performed the action, and it’s not the same kind of division that we’re used to in the physical world. It’s an active mental process. Numbers do not come “pre-divided” in the real world.

>steve patterson
who dis

>Cantor
who dat

>infinite sets are generated by the human mind
>therefore they exist

This (informal) logic indicates that infinite sets exist; not that they don't.

>formalism must adhere to out perceptions of reality
This is the dumbest shit I've ever heard.

These people know what they're talking about

This relies on having either continous times (arbitrarily many events) or continous divisivlble matter

>no infinite sets
stopped reading right there, did wildberger write this?

vocaroo.com/i/s07GzfZWgB88

No.

>continous divisivlble matter
just like in real life

Nope :^)

fuck off steve, nobody cares about your obsession with sucking wittgenstein's dick

>there are no infinite sets
>there is an axiom which states "at least one empty set exists"
Why do people not grasp the concept of foundations?

>greentexting in emails

>implying ">" notation doesn't predate chans and bbs.

Tep kok

there is no god, stop phantasising

*tips fedora*

>I understand your point. However, I cannot accomplish what you want in a blog post. First is identifying the problem. Second is presenting the alternative, which I am currently working on.
>
>If people are doing something wrong, or thinking in a logically-contradictory way, then it's valuable to point out the error, even if you don't have an alternative laid out.
>
>If somebody were building a house on top of a sinkhole, I would point it out immediately and aggressively - even if I didn't have a replacement lot for them sorted out.
>
>Best,
>Steve

PLEASE LOOK AT ME

ad.: He then asked me to give him more references to people and here is what I answered - might be of interest to others as well:

I doubt you'll find someone more willing to work towards that that Wildberger, but I can give you a few directions I deem relevant.

A more well known living historian of mathematics is Leo Corry, who notably wrote

- Modern algebra and the rise of mathematical structures (90's)

- David Hilbert and the Axiomatization of Physics (00's)

- A Brief History of Numbers (last year)

The two guys I mentioned in the last mail are the ones where I know the take a fairly dogmatic standpoint - one which I personally don't find sensible.

There are people, though, who happen to work at foundations that are neither setty nor too much informed by algebraic geometry folks related to the Bourbaki's.

Depending on how much you know about the history of formal logic, I urge you to read the two answers here* and here**. The first major step post Frege towards constructive approaches is outlined here***. And then the Russian school of constrictivism is notable.

As outlined above, that path was then overtaken by computer scientists and the bulk of mathematicans don't care. Type theorists do constructive stuff, or at least know exactly when they leave the constructive setting by adjoining non-constructive elements to their systems.

plus.google.com/ AndrejBauer/posts

Hope that helps
*
philosophy.stackexchange.com/questions/2617/how-did-first-order-logic-come-to-be-the-dominant-formal-logic
**
philosophy.stackexchange.com/questions/3318/is-first-order-logic-fol-the-only-fundamental-logic?rq=1
***
en.wikipedia.org/wiki/Brouwer–Hilbert_controversy

Holy shit that's retarded

>you can't conceive every element of an infinite set, therefore infinite sets don't exist.

I really hope he answers back with

>Physicist
>Having an opinion about mathematics
Axiom of faggotry: you can pick one and only one.

At least that is how I would respond. Get the fuck out of here.

I too wish he would have responded with meme arrows and Veeky Forums banter

>watch Wildberger vids to see what all the hype is about
holy fuck he makes rational arguments

damn when will finitists overcome this butthurt

When everyone else realizes they're wrong.
Any decade now. :^)

I don't mean to be offensive but it looks like the guy answered in pretty much the only reasonable way he could have been expected to answer. Afterwards he asked for references in order to try and understand what it is you wanted and you gave him links to babby's first hilbert-brouwer controversy.

Come on man, at least try to figure out where you disagree with his argument before you start being condescending.

>The point of the mail is debatable. I mostly want to see if he agrees that he rejects formalism in any case, in which case the case is clear.

This is what you should have asked, then.

>steve-patterson.com/cantor-wrong-no-infinite-sets/

It looks to me like his argument is similar to one of Wittgenstein's arguments. Essentially it boils down to something like
>Cantor's diagonolization argument ends in a contradiction. The usual way to resolve this is by claiming that the set is uncountable, however the correct way to resolve it is actually by realizing that infinite sets don't exist (i.e. the true contradiction lies with a different assumption further up the tree).

He seems to argue that instead one should fall back on a system of set theory/mathematics(?) analogous to computability.

I haven't read all of his work but it looks like he definitely does not believe in potential infinity (strict or otherwise).

Both of your descriptions about arithmetic are bad and you should both feel bad.

This is formally correct if you approach math from a proof theory perspective. At best one could accept countable infinity (either potential or actual).

I have a question for you.

Regarding the use of the [math]\forall[/math] quantifier, do you believe that one should be able to quantify over an infnite number of things? If so, how do you justify this?

On a sidenote, stating that there is always more is the argument of potential infinity, not the argument of actual infinity.

If all your statements and all your proofs are restricted to finite length then how can you justify the existence of actually infinite sets or worse uncountably infinite sets?

>having a literally a high school tier understanding of these concepts.

>they are not actually doing math TM
>sneaking an innocent meme arrow in the top under the guise of quoting the man

ISHYGDDT

This is truly hilarious.

>Infinite things don't exist in the real world, therefore math shouldn't deal with infinites.

Is this guy serious?

spoiler: infinity exists in real life

en.wikipedia.org/wiki/Plane_wave

>plane waves exist in real life
You haven't even baited the hook.

a few things on the op's post

part 1

>A superior response to the question, “How many positive integers are there?” is to say: “There is no inherent limitation to the size of set you can create with positive integers.”
>That doesn’t mean there’s an actually-infinite set out there in the world. It means there’s no limit to the size of the set’s construction. Contrast this to the question,
>“What is the size of the set of odd integers between 12 and 18?”
>The answer is finite. It has an inherent limitation, based on the structure of the question and the nature of numbers.

no one is saying that an actually-infinite set necessarily exists in the ‘world’. it’s an abstract concept which obviously isn’t very intuitive. interesting how the author didn’t bring up ‘existing in the world’ when talking about a finite set. the finite set doesn’t necessarily exist in the ‘world’ either
>Mathematics is a theory with its own language, but unlike most theories, it perfectly maps onto the world. In other words, mathematical principles are not a linguistic convention; they are not a hypothesis; they are derived from the laws of logic, and they apply to all universes that are composed of existent things.

math is a set of abstract rules and abstract consequences of those rules, and it’s used to describe abstract and physical things. it might perfectly represent a statement about how the physical world works, but unless author defines what he means by ‘mapping’, i’m hesitant to accept this claim.

part 2

>Mathematics produces concepts by abstracting away from concretes. We can look at a fruit bowl and say, “There are four oranges and four apples in the fruit bowl.” But we can also ask, “What is this ‘four’ property, and what does it imply?” And we can discover essential logical relationships between, say, “the amount of four” and “the amount of two” or “the amount of eight, or twelve,” etc. Regardless of whether we’re talking about four oranges, four apples, four cars, planes, or horses, the conceptual/logical relationship between different amounts is the same. It is universal.

actually no, math produces concepts by abstracting concepts from other abstract concepts. no one is looking at fruit when they discover the mathematical nature of numbers. number theory has axioms that aren’t ‘concrete’, and the consequences of these abstract axioms are abstract as well.

>modern math makes a mistake by thinking that “four” has an independent existence all by itself. That, in addition to the apples and oranges in the fruit bowl, you have “the number four” – a separate entity, that can float disconnected to any concretes.

not necessarily, using ‘four’ as a concept doesn’t mean it has ‘existence’ in the sense that the author is possibly using. if by ‘has an existence’ the author means ‘is an abstract concept made up of other abstract concepts’, then yes, ‘four’ ‘has an existence’.

part 3

>To ask, “How many positive integers are there?” is to presuppose an error. Sets aren’t “out there”. They are created. All sets are exactly as large as they’ve been created. There is no such thing as “all the positive integers”.

again, no one is saying anything about a set necessarily being ‘out there’. if an infinite set is simply an abstract idea, it doesn’t necessarily have to be out anywhere. perhaps if you think of an ‘abstract universe’ where abstract concepts ‘exist’, then perhaps something’s ‘out there’, but no one necessarily has to make an assumption about whether or not abstract universes exist before they play around with abstract concepts.

>Any number N that you conceive of, I can always think of N+1. Does that mean that N+1 exists prior to its conception

the author keeps using ‘exist’ without defining what he means. if the author’s definition of ‘existence’ doesn’t account for abstract elements that haven’t been explicitly stated, then fine. but if your definition of existence includes the set of all possible abstractions, then no.

>A set explicitly means an actual, defined collection of elements. If you ever, at any point, have an actual collection of elements, you certainly do not have an infinite amount. In order to be collected, the amount must have boundaries around it – which is an explicit denial of infinitude.

the author is perfectly fine with contemplating infinite ‘amounts’, but doesn’t tempt the idea of an infinitely large ‘boundary’. and again, this infinitely large boundary doesn’t have to be ‘out there’ in the ‘world’, it’s an abstract, nonintuitive concept.

this guy is a self absorbed joke of a human being. the about page just gave me super autism. read through it, then decide if you will really think this moron's opinion is worth anything.

>>math is a set of abstract rules and abstract consequences of those rules,
any rule is abstract, this is what ''empiricist mathematicians'' fail to get