one which to my reckoning involves substituting ∞ whenever and wherever it occurs with an appropriate finite number; this could really be a big help in my humble opinion. — Agent Smith

:lol:

:smile:

Isn't what I said implied by finitism?

↪jgill
:smile:

Isn't what I said implied by finitism? — Agent Smith

Finitism simply avoids infinity. There is no such thing as the "largest number" as far as I know.

Here's where you are going (Wiki):

Ultrafinitism (also known as ultraintuitionism) has an even more conservative attitude towards mathematical objects than finitism, and has objections to the existence of finite mathematical objects when they are too large.

I see. So I'm not exactly wrong in saying some experts object to $\infty$. Do you know any reasons why?

As per some sources, the Greeks kept $\infty$ at arm's length because of paradoxes that it generates (vide Zeno's paradoxes) and other such as x + x = x $\to$ 2x = x $\to$ 2 = 1 (:brow:)

The number that's to be swapped with $\infty$ doesn't have to be "very large" as the video on $-\frac{1}{12}$ I posted demonstrates.

a small, nevertheless most special number like −1/12
will do just fine. — Agent Smith

But, but, but... that number arises from calculating infinite sums, and explicitly not from setting a finite limit. You cannot get to it if you set a finite limit.

The common approach is to assume that any object can be divided in any way, so there is an infinity of possible divisions for each thing to be divided. In reality though, the way an object can be divided is highly dependent on the composition of the object. — Metaphysician Undercover

I would rather say:
The mathematical approach is to assume that any object can be divided in any way, so there is an infinity of possible divisions for each thing to be divided. In physics though, the way an object can be divided is highly dependent on the composition of the object.

But the problem with setting a largest number is that it rules out irrational numbers such as pi, sq-root 2 etc because they cannot continue to infinity as decimals and therefore become expressible as ratios. Of course physicists don't care about such things, and always just fudge their calculations to get roughly the right result any way, Blah blah, experimental error, uncertainty, whatever. But mathematicians have no mercy, and maths is full of irrationality ever since Pythagoras. Irrational numbers are the devil in the detail that he proved the existence of geometrically, and the fact that mathematicians (and others) are still trying to insist that maths should be fitted within the limits of their thinking is more to do with psychology than mathematics.

You could, though, go for taxicab geometry, as long as you like your circles roughly square.

But, but, but... that number arises from calculating infinite sums, and explicitly not from setting a finite limit. You cannot get to it if you set a finite limit. — unenlightened

Yea! Notice however that the $\infty$ sum has a finite value ($-\frac{1}{12}$). That's the killer move!

Notice however that the ∞ sum has a finite value (−1/12). That's the killer move! — Agent Smith

When one tries to do analytic continuation of the Riemann Zeta function where it is not warranted this kind of nonsense results. What makes it useful in physics is beyond me.

The mathematical approach is to assume that any object can be divided in any way, so there is an infinity of possible divisions for each thing to be divided. In physics though, the way an object can be divided is highly dependent on the composition of the object. — unenlightened

Yes, in physics the way an object "can be divided" is highly dependent on the composition of the object, but sometimes this fact is ignored by physicists. That's the problem I referred to with the way that frequencies and wavelengths are treated. The problem appears to be that there is no identified medium within which the electromagnetic waves exist, so the real properties of the waves cannot be determined or described, and physicists are left with theoretical mathematics which assumes an infinity of possible divisions.

But mathematicians have no mercy, and maths is full of irrationality ever since Pythagoras. Irrational numbers are the devil in the detail that he proved the existence of geometrically, and the fact that mathematicians (and others) are still trying to insist that maths should be fitted within the limits of their thinking is more to do with psychology than mathematics. — unenlightened

I would say that if the geometry can prove the existence of irrational numbers, then this is an indication that the geometry is faulty. The issue with pi and the square root of two appears to involve the traditional geometer's use of distinct dimensions. So the irrational nature of the square root of two shows that any two distinct dimensions, produced by the right angle, are fundamentally incompatible. Set two new points, equidistant from a starting point, on two traditional dimensions (making a right angle), and the distance between those two points will always be indefinite (an irrational number).

The issue with pi is very similar. A straight line has one dimension, and a curved line has two dimensions. These two types of lines are fundamentally incompatible. Some might say that an infinitely large circle, has an arc so gradual that it's actually a straight line, but how would you ever get two dimensions, a curve, or a circle then? Likewise, some might say that a polygon with infinite sides is actually a circle, but this doesn't really reconcile the incompatibility, because each side is still a one dimensional line, at an angle to the next side, it's sides are distinct straight lines and that's not a curved line.

I like your example of the taxicab geometry. I'm not sure, but it appears to deny the reality of the one dimensional straight line. But it appears to me, like if we deny the straight line, then we'll need an infinite number of dimensions, because there would be an infinite number of possible ways to get from one point to another point. So this doesn't really get rid of the irrationality.

When one tries to do analytic continuation of the Riemann Zeta function where it is not warranted this kind of nonsense results. What makes it useful in physics is beyond me. — jgill

This $(\sum_{n=1}^{\infty} = -\frac{1}{12})$"nonsense result" has applications in science, in string theory to be precise. I thought mathematicians like yourself would know. Oh, but I repeat myself. Apologies.

You seem fascinated by this anomaly. It's a kind of summation result for series that don't converge in the mathematically acceptable manner. Here's a link that should keep you occupied until bedtime:

1 + 2 + 3 + ...

You seem fascinated by this anomaly. It's a kind of summation result for series that don't converge in the mathematically acceptable manner. Here's a link that should keep you occupied until bedtime:

1 + 2 + 3 + ... — jgill

Well, it is fascinating is it not? If we could somehow prove that $-\frac{1}{12}$ can take the place of $\infty$ in every calculation $\infty$ pops up in, we've struck gold, oui monsieur?

we've struck gold, oui monsieur? — Agent Smith

Fool's gold.

Fool's gold. — jgill

Why? Are you saying string theory and all the other physics topic in which $\infty$ is swapped for $-\frac{1}{12}$ is nonsense?

Is mathematics inconsistent?

$(\sum_{n=1}^{\infty} = \infty) \land (\sum_{n=1}^{\infty} = -\frac{1}{12})$

Is mathematics inconsistent? — Agent Smith

The expression on the right is a way of summing a divergent series. (assuming we know what you mean with the notation)

The expression on the right is a way of summing a divergent series. — jgill

You're not the spoon feeding kind are you? :up:

You got me wondering if anyone does this line of research anymore. My conclusion, rarely. Here are a couple of papers, the second being more a survey.

Summing Series

Summing Series

Most kind of you! Gracias jgill, gracias.

x + x = x → 2x = x → 2 = 1 — Agent Smith

For finite cardinals, if ~x = 0, then ~x+x = x.

For infinite cardinals, x+x = x and 2x = x.

But you cannot infer 2 =1 from 2x = x when x is an infinite cardinal. Cardinal addition where one of the multipliers is an infinite cardinal does not have the cancellation property.

Once again, your ignorance and intellectual dishonesty have enabled you to post a false claim.

Is mathematics inconsistent? — Agent Smith

No derivation of a contradiction has been shown in ZFC.

And you write (I'm using plain text):

Sum[n = 1 to inf] = inf
Sum[n = 1 to inf] = -(1/12)

As far as I can tell, those are not even well formed.

Sum[n=0 to inf] requires a term on its right*, otherwise it's just a dangling variable binding operator.

* E.g., Sum[n=0 to inf] 1/(2^n) is well formed and meaningful.

Please edify me then. How does intuition work in math? How is it related to so-called mathematical/logical rigor? Talking to you is like conversing with a computer. DOES NOT COMPUTE! DOES NOT COMPUTE! From start to finish, that's all you say! I should call tech support! — Agent Smith

There's lot to unpack there.

(1) (a minor point) You were responding to a quote of mine about intuitionism. That's the wrong quote, since the subjects of intuition and intuitionism are related but very different subjects. Instead the pertinent quote is "I quite understand that human thinking, including about mathematics, involves intuition. Indeed I'm interested in the relation between formal theories and intuitions."

(2) "How does intuition work in math?" The subject of intuition and mathematics is a big one. I could not even summarize my thoughts about it in an ad hoc post. And I do not have conclusive things to say about it. I said that I think about the subject a lot; but I do not claim to have arrived at firm conclusions about it.

(3) The subject of the relation between intuition and rigor is narrower than just mentioned, but still a big one. It's not clear to me where the best place to start would be. But perhaps one area to begin is the notion that formalization in its most basic intuitive sense can, in principle, be reduced to series of discrete observations about discrete objects conceptualized as being indivisible, such as, witnessed in physical form, tally marks on paper or 0s and 1s on paper. My own personal imagination doesn't provide that there could be any form of participation in mathematical reasoning more basic.

(4) Even if it seems to you that my postings read as mechanical or computer-like, that is not a refutation of anything I've posted. Moreover, it would be a non sequitur to infer that I don't think about the subject of intuition in mathematics from your premise that my writing is computer-like. Moreover, as to writing style, one should take into account that my purpose is not to entertain you, nor to present to you as loosely gabbing. When you post plainly incorrect claims about mathematics, then usually my main purpose is to clearly point out that you are incorrect and often to explain why; and often that is best achieved by explanations that use uniform terminology and parallel forms.

(5) You are lying. It is not even remotely true that I only say that you are incorrect. Over many months, I have also given you quite generous and detailed explanations why you are incorrect. It is remarkably dishonest and boorish of you to say otherwise.

to say that there is a collection of objects with no objects is contradictory — Metaphysician Undercover

You skipped my previous remarks about that.

you are now trying to reduce many to one, by saying that a set is an object. — Metaphysician Undercover

If I recall, we discussed this at length many months ago. Again, I don't mean a physical object. I mean a set is an abstract object.

If you don't accept the notion of abstract objects, then I admit that there's not much for us to discuss. I do not feign to be able to explicate the notions of 'abstract' and 'object' beyond ordinary understanding of such basic rubrics of thought as acquired ostensively or by whatever means people ordinarily understand them.

If you do accept the notion of abstract objects, then I point out that a set theoretical intuition may begin with the notion of a thing being a member of another thing: The notion of membership. First, that obviates the need even to use the word or notion 'set', as instead we merely discuss the membership relation. Second, even though the word or notion 'set' is dispensable, we can still go on to define it. Also, in the formal theory itself, the word or notion 'object' does not occur, though when we informally talk about informal theories it would be cumbersome to eschew the word and notion 'object'. In that context, mathematicians readily understand that a set is an abstract object. There is nothing in the definition of the word 'object' that precludes an abstract object nor that precludes that certain abstract objects, viz. sets, are in relation to others, viz. membership.

students are taught very specific principles, and their minds are funneled down a very narrow path — Metaphysician Undercover

That offers at least these prongs of refutation:

(1) I am mostly (but not exclusively) self-taught from textbooks; and textbooks in mathematics don't indoctrinate. Rather they put forth the way the mathematics works in a context such as presented in the book. A framework is presented and then developed. There is no exhortation for one to believe that the framework is the only one acceptable.

(2) Indeed, mathematics, especially mathematical logic, offers a vast array of alternative frameworks, not just the classical framework, including constructivism, intuitionism, finitism, paraconsistency, relevance logic, intensional logic .... And mathematics itself does not assert any particular philosophy about itself, as one is free to study mathematics with whatever philosophy or lack of philosophy one wants to bring to it.

(3) It is actually cranks who are narrowminded and dogmatic. The crank insists that only his philosophy and notions about mathematics are correct and that all the mathematicians meanwhile are incorrect. The crank doesn't even know anything about the mathematics yet the crank is full of sweeping denunciations of it. The crank makes wildly false claims about mathematics, and then doesn't understand that when he is corrected about those claims, the corrections are not an insistence that the crank agree with the mathematics but rather that the corrections merely point out and explain why what the cranks says about mathematics is untrue. It's as if the crank says, "classical music is all wrong because classical music never has regular meter" and then when it is pointed out that most of classical music does have regular meter, the crank takes that as narrow minded demand that he like classical music. And the crank is not even aware that mathematics, especially mathematical logic, offers a vast array of alternatives. Meanwhile, the crank's usual modus operandi is to either skip, misconstrue, or strawman the refutations and explanations given to him, thus an unending loop with the crank clinging to ignorance, confusion, and sophistry.

The students are discouraged from looking outside the program — Metaphysician Undercover

Does Metaphysician Undercover have actual incidents to cite? Is there a particular incident to which he is witness, or widespread reports of them that would justify such a sweeping generalization or even a more modestly limited claim?

TIDF split the scene — Metaphysician Undercover

Wrong again.

If you don't accept the notion of abstract objects, then I admit that there's not much for us to discuss. — TonesInDeepFreeze

I cannot agree that abstractions are objects, unless we restrict "object" to refer only to abstractions. But then we could not use "object" to refer to anything else, or we'd have equivocation. And we would have to create a special form of the law of identity, such that when 'the same' abstraction exists in the minds of different people, we can still refer to it as "the same" abstraction, despite accidental differences between one person and another, due to different interpretations. The current law of identity requires that accidental differences would constitute distinct 'objects' which are therefore not the same, so we'd need a different law of identity.

If you do accept the notion of abstract objects... — TonesInDeepFreeze

If the law of identity describes what an "object" is, then I cannot accept the notion of abstract objects, for the reasons explained above. I might accept the notion of abstract objects, so long as you agree with me that the law of identity does not apply to this type of object, and we proceed with caution, so as not to equivocate between these two distinct types of objects, physical objects being described by the law of identity, and abstract objects being a different type of object to which the law of identity does not apply. Can you agree to that?

then I point out that a set theoretical intuition may begin with the notion of a thing being a member of another thing: The notion of membership. — TonesInDeepFreeze

Sure, "a thing" in this context, I assume is an abstract object, with no possibility for identity, so you can say whatever you want about "a thing", or "object". You can even say contradictory things about an object, because the object has no identity as a physical object does. You could say one thing about "the object" and I could say a contradictory thing about it, because there's nothing to ensure that we maintain consistency between the object in your mind, and the object in my mind, which both bear the same name. How would we even know that we're talking about the same object, except that we are using the same name? Do you propose that the name is the object? Then where is the abstraction?

So there are no restrictions in the sense of truth or falsity by correspondence, and to say contradictory things about the same object might be completely acceptable. The "thing" is a purely imaginary fiction, and we can use contradiction in fiction without a problem, though it might make the imaginary "thing" seem counterintuitive.

The notion of membership. — TonesInDeepFreeze

All right then, considering the above conditions, I'm ready to try and understand what "membership" means. What does it mean for one thing to be a member of another thing? Is it necessary that the thing which is a member be a different type of thing from the thing which it is a member of? If not, how would I distinguish between which things are the partakers, and which are partaken of?

That offers at least these prongs of refutation:

(1) I am mostly (but not exclusively) self-taught from textbooks; and textbooks in mathematics don't indoctrinate. Rather they put forth the way the mathematics works in a context such as presented in the book. A framework is presented and then developed. There is no exhortation for one to believe that the framework is the only one acceptable.

(2) Indeed, mathematics, especially mathematical logic, offers a vast array of alternative frameworks, not just the classical framework, including constructivism, intuitionism, finitism, paraconsistency, relevance logic, intensional logic .... And mathematics itself does not assert any particular philosophy about itself, as one is free to study mathematics with whatever philosophy or lack of philosophy one wants to bring to it.

(3) It is actually cranks who are narrowminded and dogmatic. The crank insists that only his philosophy and notions about mathematics are correct and that all the mathematicians meanwhile are incorrect. The crank doesn't even know anything about the mathematics yet the crank is full of sweeping denunciations of it. The crank makes wildly false claims about mathematics, and then doesn't understand that when he is corrected about those claims, the corrections are not an insistence that the crank agree with the mathematics but rather that the corrections merely point out and explain why what the cranks says about mathematics is untrue. It's as if the crank says, "classical music is all wrong because classical music never has regular meter" and then when it is pointed out that most of classical music does have regular meter, the crank takes that as narrow minded demand that he like classical music. And the crank is not even aware that mathematics, especially mathematical logic, offers a vast array of alternatives. Meanwhile, the crank's usual modus operandi is to either skip, misconstrue, or strawman the refutations and explanations given to him, thus an unending loop with the crank clinging to ignorance, confusion, and sophistry. — TonesInDeepFreeze

I disagree on a number of key premises here, so I do not consider any of this to be acceptable refutation. But that's all beside the point, just a difference of opinion on trivial matters.

Wrong again. — TonesInDeepFreeze

I'm glad to be wrong, of course. I actually enjoy discourse with you TIDF, you're generally well behaved and intelligent. I just don't see that you know any strong principles.

:smile:

My Thoughts on Intuition & Logical rigor in Math

Intuition is more a feeling than a thought. I've experienced it in doing high school math. There was this time when I was solving a problem in an exam and I soon realized the numbers mid-calculation were just too large & awkward, they didn't feel right to me; I went back to check my work and found out I had made a boo-boo. This is what I mean by intuition in general and mathematical intuition in particular.

Too, last I checked, mathematicians love to guess, formally termed conjectures. As far as I can tell, mathematical conjectures are intuitions and not just wild guesses - the former tend to be hard to prove/disprove while the latter are easily tackled by even amateurs.

That's all she wrote!

Once again, your ignorance and intellectual dishonesty have enabled you to post a false claim. — TonesInDeepFreeze

So those who're learning are guilty of intellectual dishonesty? Gimme a break!

No derivation of a contradiction has been shown in ZFC.

And you write (I'm using plain text):

Sum[n = 1 to inf] = inf
Sum[n = 1 to inf] = -(1/12)

As far as I can tell, those are not even well formed.

Sum[n=0 to inf] requires a term on its right*, otherwise it's just a dangling variable binding operator.

* E.g., Sum[n=0 to inf] 1/(2^n) is well formed and meaningful. — TonesInDeepFreeze

:chin:

Is there a finite number (Nmax) such that no calculations ever in physics will exceed that number? — Agent Smith

Yep. The maximum entropy or information content of the visible universe is 10^123 k. So if you wanted to number every individual degree of freedom that exists for all practical purposes, there’s your number.

At least it is the current best stab. See https://mdpi-res.com/d_attachment/proceedings/proceedings-46-00011/article_deploy/proceedings-46-00011-v2.pdf?version=1612356633

Merci, merci apokrisis.

$10^{123}$

:fire: