## Saturday, December 6, 2014

### Reality of Numbers, but Not of Numeric Infinity

1) ... on reality of existence of numbers (and on Pythagoreans and Bruno), 2) Reality of Numbers, but Not of Numeric Infinity, 3) Jamma starts giving examples! Yeah!

Video commented on
Infinite Fractions - Numberphile
Numberphile

Hans-Georg Lundahl
"Infinite Fractions" - no such thing.

Potentially infinite serial fractions [what I meant is called "continued fractions], like ways of writing π as a sequence, yes. But are never actually executed.

And the relation of perimeter to diameter is not per se infinite, just not adequately numerisable for its accurate value.

For an approx value, that's another matter.

jamma246
So is there such thing as a fraction?

Honestly, these pseudo-philosophical arguments about what kinds of mathematical objects exist and which don't always end up being silly.﻿

sibtain ali
There are infinite numbers so infinite fractions﻿

Hans-Georg Lundahl
A finite fraction very certainly exists, like 1/3 or 2/5. It is very certainly also executed.

Certain ratios also exists which could be reinterpreted as infinite series of serial fractioning. but those serial fractionings are never executed any more than a geometer ever executes "take a line of infinite length".

So, no, what kinds of mathematical objects exist and what kinds do not exist is a very legitimate philosophical - not pseudophilosophical but really philosophical question which very much interests me.

Oh, by the way, the kind of ratios that cannot be expressed accurately and exactly as finite fractions but only as infinite serial fractions, like π, are never number-to-number ratios but more like domain of size-to-size ratios or similar mathematical continua outside arithmetic proper.

sibtain ali, no, there is no such thing as a number which is infinity and there is no such thing as a fraction which contains fraction within fraction up to infinity being reached.

The series of different numbers like 1 2 3 4 and of different fractions like 1/2, 2/2, 3/2, 4/2 ... 1/3, 2/3, 3/3, 4/3 and so on are potentially infinite in so far as mathematicians cannot know when the next number or possibly even the next fraction ceases to count or account for something in the universe in its mathematical aspect.

jamma246
"A finite fraction very certainly exists, like 1/3 or 2/5. It is very certainly also executed."

I'm sorry, I have no idea what you are talking about. What do you mean by "executed"? What is your criterion for a certain mathematical object "existing", what does that even mean?

Does the square-root of 2 make it onto your list of mathematical idealisations which "exists"? Does the set of natural numbers "exist"?

Honestly, what you are saying is total nonsense:

"The series of different numbers like 1 2 3 4 and of different fractions like 1/2, 2/2, 3/2, 4/2 ... 1/3, 2/3, 3/3, 4/3 and so on are potentially infinite in so far as mathematicians cannot know when the next number or possibly even the next fraction ceases to count or account for something in the universe in its mathematical aspect."

There is no claim that this quantity "accounts for something in the universe"; how would one even define what that means? That seems ridiculously subjective for me, and yet you speak about it with such authority.﻿

Hans-Georg Lundahl
"What do you mean by "executed"? What is your criterion for a certain mathematical object "existing", what does that even mean?"

I mean that the mathematician overlooks every "number" of the fraction and its fractionality.

It is executed by the fact of writing it. [By writing it out in full.]

While a serial fraction is never executed, never written out in full, never given a fully accurate simplification if it is "infinite".

That is what I mean.﻿

"Does the square-root of 2 make it onto your list of mathematical idealisations which "exists"?"

As a piece of geometry, yes.

As a piece of arithmetic, no.

It is comprehended as for instance "hypotenuse of any right angled triangle of which both kathetoi (not sure what that is in English) are same length, relative to that length taken as one".

It is not a numerical execution, but a comprehensible one. Of a piece of geometry.﻿

"There is no claim that this quantity "accounts for something in the universe"; how would one even define what that means?"

I might say I am not Kantian. I do not believe the universe is infinite.

Number of atoms is limited.

Number of conglomerates of atoms is limited.

Number of events are limited.

Some numbers handled theoretically in Arabic numerals and perhaps with a "*10n" may be beyond any finite number of finite things or events to count.

But the meaning in which numbers are "potentially infinite" is that we have no mathematical way in which to decide what that greatest de facto real number would be.﻿

Hans-Georg Lundahl
The only sense in which I am "subjective" is the fact that I refuse to be intersubjective with people attributing infinity to the creation and doing that in mathematcial ways.

I am no more subjective than any other serious believing Catholic.﻿

jamma246
The philosophy of mathematics is an interesting subject, but many of these observations have already been answered and, to be honest, you are speaking a load of gibberish.

There are certainly interesting foundational questions; what things can and cannot be proved in mathematics, which constructions are or are not permitted... given the initial axioms!. For example, it is perfectly acceptable to say that one has different results or methods depending on whether or not one assumes the truth of the Axiom of Choice. Constructivist mathematics has a different flavour to non-constructivist mathematics. But it is pointless to ask whether or not the AoC is true or "exists". It is simply an axiom. Similarly, it is meaningless to declare some mathematical object or another as "existing" in a real sense. All mathematical constructions are idealised constructs, by definition.

We've moved past the time where people would shun, for example, the complex numbers for not being "real". Our understanding is now far more sophisticated than that. Number 2 is no more "real" than π or sqrt(-1) is, but one may still work with initial axioms which do or do not permit certain constructions. There is great generality with which one can extend these ideas, see for example Topos Theory.﻿

[2 and π are both real. But in diverse disciplines, see a previous commentary. But sqrt(-1) and even "-1" if taken as a number rather than as a "relative number" or "numeric relation", if taken as "sth less than zero" is not. Nor is zero.]

Hans-Georg Lundahl
My axioms and the common sense ones are:

• number means either one single or one single several times over.

Start arithmetic from there;

• size relations can be like relations between numbers (e g number 3, 4, 5 can have the same relation between them as the 3, 4, 5 sides of the Egyptian triangle);

• size relations can also be NOT like relations between any numbers (e g a triangle with two sides equalling each other and having a right angle could call that equality a 1:1, a 2:2, a 3:3 equality, whichever they liked, but either way the relation of the third side, of the hypotenuse, to each of the others would be sqrt(2) which is simply NOT a number).

Geometry needs more axioms than these to work, but these are two basic ones so as not to confuse it with nor over separate it from arithmetic.

Unlike arbitrarily chosen axioms, these are rooted in the nature of the kind of things that mathematics actually studies. Or used to study.﻿

Hans-Georg Lundahl
After actually watching the video:

• a man can go on with the Stern Brocot sequence and the fractions it generates as long as he likes or can manage, but that is a VERY finite amount which is executed, like on the triangular form he executed down to x/4 and x/5 but no further;

• God can execute all of the sequence at once - but He has also chosen how much of it which corresponds to meaningful and correct information about any relations of quantities in the universe and which other ones do not.

I thought we were dealing with something else, a but more complicated.

π Continued Fraction
http://mathworld.wolfram.com/PiContinuedFraction.html

Continued Fraction
http://mathworld.wolfram.com/SimpleContinuedFraction.html

In these case each fraction itself by being infinite is never fully executed.﻿

jamma246

Hans-Georg Lundahl
People who are as uninterested in philosophy as you might do well to leave maths alone as well.

But I am stopping as I have made my point.﻿

jamma246
I can assure you that people in mathematics often love philosophy. But they don't enjoy meaningless pseudo-philosophical ramblings.﻿

Hans-Georg Lundahl
Nothing in what I said was pseudo-philosophical and nothing in what I said was meaningless.

codediporpal
The universe disagrees. Just because you can't compute it doesn't mean the universe isn't doing it all the time.

jamma246
Hi[s] reasons depend on dogmatism and "God". I wouldn't bother trying to argue with it - meaningless waffle, the antithesis of mathematics.

Hans-Georg Lundahl
Oh, no.

An "infinite distance" is for example an oxymoron.

Any distance is between two points, any point at the end of a distance ends it there, any distance ended in two points is finite.

So, no such thing as an infinite distance.

So, I have full rational backing for my observation - it's you atheists who have none.

Summa Theologica I P, Q7, A3
Article 3. Whether an actually infinite magnitude can exist?

"Objection 1. It seems that there can be something actually infinite in magnitude. For in mathematics there is no error, since "there is no lie in things abstract," as the Philosopher says (Phys. ii). But mathematics uses the infinite in magnitude; thus, the geometrician in his demonstrations says, "Let this line be infinite." Therefore it is not impossible for a thing to be infinite in magnitude."

And skimming over rest of objections and skimming over his systematic answer we get the answer to this one:

"Reply to Objection 1. A geometrician does not need to assume a line actually infinite, but takes some actually finite line, from which he subtracts whatever he finds necessary; which line he calls infinite."

The mathematician in this video proceeded no differently. He only executed the sequence to a not just finite but even very low magnitude. We were only dealing with x/4 and x/5 here.

The point is not just that these are finite, but that however much greater he had executed it to, he would still have executed only a finite sequence and its last items would still be finite.

Here is the whole article, and note the next one:

First Part Q7. The infinity of God
Article 3. Whether an actually infinite magnitude can exist?

Ibidem : Article 4. Whether an infinite multitude can exist?

The next one concerns us even more. There is no such thing as an actually infinite multitude.

jamma246
Hahaha, from 'The Summa Theologica of St. Thomas Aquinas'.﻿

Well, have fun continuing to base your philosophies on your religion. In the meantime, mathematicians and scientists will continue making fundamental and important contributions to the real world, using actual mathematics, which can perfectly well cope with infinities, and has done for centuries.

There's a reason why logic and science always say a lot more about the real world, and are a lot more useful than philosophical prejudice.

Hans-Georg Lundahl
"Hahaha, from 'The Summa Theologica of St. Thomas Aquinas'."

I quite agree, except on the "hahaha" part.

I am as said not BASING my philosophy on my religion.

I am ACCEPTING religion as a corrective to philosophy whenever faulty, but in this case, the case for the Thomistic position is very clear independently of Catholicism.

"In the meantime, mathematicians and scientists will continue making fundamental and important contributions to the real world,"

As a Thomist, I like observing and understanding it correctly before making a contribution to it.

"using actual mathematics"

So did St Thomas. Here.

"which can perfectly well cope with infinities, and has done for centuries"

"Cope with infinities" is something other than using infinity, other than as a concept verging on fiction.

St Thomas' example with the Geometrician whose "infinite line" is just a line from which he can deduce as much as he needs is pretty close to what mathematicians do to this very day.

"There's a reason why logic and science always say a lot more about the real world, and are a lot more useful than philosophical prejudice."

That's a nice philosophical prejudice on your side!

Now, philosophy and science are normally coterminous. There is a correct and logical way to do it, like St Thomas did, and there is a prejudiced way, like you do.

Mathematicians have and use some kind of concept of "infinity" - your prejudice is that their terminology is correct and complete. My observation is one would at least need to add "potential" to "infinity" before it makes any sense.

jamma246
Well, I'm sorry, but the ideas of Thomas Aquinas have been almost wholly demolished by this point. He had nothing useful to say about reality whatsoever. Welcome to the more enlightened 21st century.

He was clearly an intelligent person, but his religious convictions lead him to naive and unreasoned statements about reality. It is no coincidence that when we stopped pretending that mathematical concepts were physical ones, the subject leapt forward and we started making real progress. It doesn't matter whether negative numbers 'exist' or complex numbers 'exist' - they are useful and well-defined tools (and asking whether such things 'exist' is meaningless anyway). The same goes for infinity.

Can you tell me any actual contributions that philosophers such as Thomas Aquainas made to science or mathematics? Anything to back up the claim that they saw the world more clearly than a pure mathematician such as, say, Grothendieck? What useful mathematics did Thomas Aquinas ever come up with? The mathematicians with fewer prejudices (such as against infinity) who saw the subject for what it is, as abstraction, were the ones who proved actual rigorous results with real world consequences. Take any list of the greatest mathematicians of the last thousand years, and they wouldn't have had these silly prejudices against mathematical concepts just because they couldn't cook up something from the real world which seemed to resemeble the mathematical concept. There were a couple of construtivists, such as Brouwer, but most of these seemed to understand that doing constructivist mathematics is simply to study a different branch of mathematics, where certain axioms aren't assumed, rather than to claim that the other approach was 'wrong'.

Infinity pervades mathematics. Even in subjects such as combinatorics it is useful to use infinity. It is simply a tool, just as any concept of mathematics is, and is as well-defined as any other mathematical concept. Talk to actual mathematicians and they will have no idea what your "potential" infinities are, and these are the people who understand infinity most clearly. You are talking a load of nonrigorous nonsense, in a subject that is now easily understood for someone who wants to take the time to study it.

Your line, for example, infinite in length or not. How many points does it contain? Do right-angled triangles 'exist'? Does a right-angled triangle with two sides of length 1 exist? If so, then irrational lengths exist, since the square root of two does. But to express the square root of two, one needs something like a non-periodic infinite decimal, or an eventually periodic infinite continued fraction. Do you have any reason for thinking that space is only made up of finitely many points? Even if it is, do you not think that it is useful to approximate it by something of infinitely many points?

All of these silly points can be avoided by simply not holding to these petty convictions about what does and doesn't 'exist' in mathematics, and instead take the modern approach. Mathematics is about axioms and deductions that follow from them, which in themselves are subject to the reality of logic. Not placing arbitrary restrictions is the best way. Your ideas are 10 centuries out of date.

Hans-Georg Lundahl
"He was clearly an intelligent person, but his religious convictions lead him to naive and unreasoned statements about reality"

"Naive" perhaps to the over sophisticated. Unreasoned? Never.

"It is no coincidence that when we stopped pretending that mathematical concepts were physical ones, the subject leapt forward and we started making real progress."

In making more and more complex calculations, yes.

In understanding them properly - definitely no.

"It doesn't matter whether negative numbers 'exist' or complex numbers 'exist' - they are useful and well-defined tools (and asking whether such things 'exist' is meaningless anyway). The same goes for infinity."

It matters for our understanding.

I am not denying they are useful as fictions. So is reading Tolkien. Or C. S. Lewis. Some might even cite Isaac Asimov, though I wouldn't. But do not smudge out the difference between mathematical realities, which are one side of physical realities, and mathematicians' fictions.

"Can you tell me any actual contributions that philosophers such as Thomas Aquainas made to science or mathematics?"

His doctor while a student at Sorbonne, St Albert, was a great zoologist, and laid grounds for geology and palaeontology.

The latter's philosophical opponent Roger Bacon gave us eye-glasses and laid the grounds for optics.

In geology St Albert was not really superseded until Steno, in optics Roger was not really superseded till Newton.

"Anything to back up the claim that they saw the world more clearly than a pure mathematician such as, say, Grothendieck? What useful mathematics did Thomas Aquinas ever come up with?"

What use have we had of Grothendieck's mathematics, when it comes to that?

"The mathematicians with fewer prejudices (such as against infinity) who saw the subject for what it is, as abstraction,"

Abstraction and fiction are different concepts.

Three is an abstraction from such things as three apples, three Graces, three goddesses, three dimensions, three parts of time, three parts of mind (memory, understanding, will), Holy Trinity - some of which are fictional (not the last one).

Infinity is - in mathematics - either a short way of saying "potentially infinite", as St Thomas took it, or, when forgetting that, a fiction, the usefulness of which varies depending on context.

In pixar repeating a smoothing out "to infinity" means repeating it till it looks smooth. A fractal is never drawn out to infinity. Neither by computer nor by anyone else - the zooming in to thousand times smaller at the same time zooms out the original picture, every version of a fractal is finite and operated by a finite number of steps by the computer.

In understanding of the universe "infinity" played as bad tricks on Kant as "zero" and "minus" is playing on Krauss. Prenamed Lawrence. Famous for being part of film The Principle, for saying things like "a star died so that you can live", or for calling the universe a "quantum fluctuation in absolute nothing".

Zero is a useful fiction even with a fictiionally negative side on the other side of the positive side, but taking that sense of the word and mixing it with the other meaning as "nothing", like Krauss does, is catastrophic.

"were the ones who proved actual rigorous results with real world consequences."

A fiction can have real world consequences.

JRRT's fictional characters have lots of real world consequences in paper, ink, films by Peter Jackson, students of Quenya, live role players and - partly at least - in a new understanding of communal ethics and in some cases conversion to Catholicism. Not to mention giving me a hint of angelic movers theory also found in St THomas Aquinas.

"Take any list of the greatest mathematicians of the last thousand years, and they wouldn't have had these silly prejudices against mathematical concepts just because they couldn't cook up something from the real world which seemed to resemeble the mathematical concept."

That would be 1014 to 2014 ... the list would include at least Thomas Bradwardine, a scholastic who stumbled on the concept of logarithms, i e of geometrical ratios (logoi) that shadow numbers (arithmoi), when discussing physics. Obviously he wasn't the one who worked it out.

Zero was not accepted as anything but a fiction until 1500-1600 sth.

"Talk to actual mathematicians and they will have no idea what your "potential" infinities are, and these are the people who understand infinity most clearly"

They consistently use it as meaning a potential or a fiction. Even when not using the terminology.

And St Thomas understood infinity clearer.

"Your line, for example, infinite in length or not."

Only finite is possible. Any line between two points is ended in both ends. I e finite.

"How many points does it contain?"

A malformed question, or the answer is, so far, two.

If you draw a midpoint, it has three actual points. And so on.

There are infinitely many points at which one CAN divide the line, so the number of points is POTENTIALLY infinite. But there are not infinitely many points at which it is actually DIVIDED, so the number of points is not infinite.

Any of the "infinite number of points" between the end points would simply be a kind of endpoint to an "infinite number" of other lines, smaller than the one we are looking at. But these other lines are not as yet realised any of them, so these infinities are just ... potential. Again, some of these potentialities can be realised. You can draw a midpoint, for instance. That augments the number of actual points to three and the number of actual lines to three also - the whole, the one part on one side of mid point, the other part on other side of mid point.

"Do right-angled triangles 'exist'?"

Certainly. SOME of them even, exotically or emblematically enough for geometry, have size relations that MIMIC number relations.

Like the Egyptian triangle.

But 3:4:5 means one thing when applied to three apples, four apples, five apples and another thing when applied to the Egyptian triangle. In one case we talk about number, in another case about relative magnitude.

Magnitude and multitude are not the same.

"Does a right-angled triangle with two sides of length 1 exist?"

Indeed, and less exceptionally this involves a size relation which cannot be parallelled in number relations.

Triangle sides AB=3, BC=4, CD=5 has for each side an exact parallel in apples.

Triangle sides AB=1, BC=1, CD=sqrt(2) have a size relation which functions only as a size relation, but not when counting apples. That is true for any "irrational number" and that is exactly how they were treated in the Middle Ages. As belonging to geometry but not to arithmetic.

Same is true of π.

"If so, then irrational lengths exist, since the square root of two does."

Lengths always have ratios to each other, it is just that not all of these mimic number to number ratios.

"But to express the square root of two, one needs something like a non-periodic infinite decimal, or an eventually periodic infinite continued fraction."

To express the sqrt of two you can write sqrt(2).

Any decimal expression is an approximation forever inexact of this in successive numeric ratios.

In itself the lengths are not numbers and their ratio is no number.

"Do you have any reason for thinking that space is only made up of finitely many points?"

Space is NOT actually made up of points at all. Any actual point is a division of a line, any actual line of a surface, any actual surface of a body.

Remember the line with only two points? One end point - other end point.

"Even if it is, do you not think that it is useful to approximate it by something of infinitely many points?"

Approximations are fictions. Very useful fictions in some contexts, but fictions.

Sqrt(2) is an exact expression. 1.414 is an approximation. Useful enough for calculations (how much paper do I need for a square with twice the surface - multiply by 1.414), but less useful for understanding what sqrt(2), since giving a false impression it is kind of a number.

"All of these silly points can be avoided by simply not holding to these petty convictions about what does and doesn't 'exist' in mathematics, and instead take the modern approach."

The points are not silly.

The answers are instructive - if taking the Thomistic approach.

jamma246
Exactly, if taking the Thomistic approach. Which has shown itself to prove useless in talking about mathematics.﻿

Hans-Georg Lundahl
As said, take the Thomistic approach and you get instructive answers. Take your approach and you found the questions or points silly yourself. In other words, your modern approach was not able to get an instructive answer out of them, it is your modern approach which has proven useless, not in applying mathematics, or rather even there, since applications don't strictly depend on taking it, but very much in talking about it, in doing the philosophy of mathematics.