Infinities

[Alan Grayson]

If there are infinities in mathematics, but not in physics or in nature, is that a problem? AG

[Philip Thrift]

On Thursday, August 29, 2019 at 9:33:23 PM UTC-5, Alan Grayson wrote:

If there are infinities in mathematics, but not in physics or in nature, is that a problem? AG

Responses to this question might be entertaining.

But the best answer is here:

       https://plato.stanford.edu/entries/fictionalism-mathematics/

       https://www.iep.utm.edu/mathfict/

@philipthrift

[smitra]

On 30-08-2019 04:33, Alan Grayson wrote:
> If there are infinities in mathematics, but not in physics or in
> nature, is that a problem? AG

https://www.youtube.com/watch?v=U75S_ZvnWNk

Saibal

[Lawrence Crowell]

On Thursday, August 29, 2019 at 9:33:23 PM UTC-5, Alan Grayson wrote:

If there are infinities in mathematics, but not in physics or in nature, is that a problem? AG

Infinity is not a number in the standard sense. It is a measure of a set, or cardinality that does not obey ordinary arithmetic. Cantor showed there exists a hierarchy of transfinite numbers and they obey a strange arithmetic such as aleph_m + aleph_m = aleph_m. In physics we do not expect to observe or measure infinity quantities. Even with Ohm's law V = IR for a given voltage potential V across a zero resistance material we predict an infinite current. However, a very low resistance material with a huge current physically adjusts itself, maybe by heating up, to increase the resistance. Zero resistance corresponds by reciprocal relationship an infinite conductivity, and this will lead to the observation of infinite currents. By way of contrast an infinite resistance means zero conductivity, and your VOM meter will show infinite Ohms, but really this is zero conductivity. We do not measure infinite quantities that are dynamical.

LC

 

[Philip Thrift]

You might be interested in

Omega++: Certified Reasoning with Infinity

http://loris-5.d2.comp.nus.edu.sg/SLPAInf/

We demonstrate how infinities improve the expressivity, power, readability, conciseness, and compositionality of a program logic. We prove that adding infinities to Presburger arithmetic enables these improvements without sacrificing decidability. We develop Omega++, a Coq-certified decision procedure for Presburger arithmetic with infinity and benchmark its performance. Both the program and proof of Omega++ are parameterized over user-selected semantics for the indeterminate terms (such as 0 * \inf).

@philipthrift

 

[Bruno Marchal]

On 30 Aug 2019, at 04:33, Alan Grayson <agrays...@gmail.com> wrote:

If there are infinities in mathematics, but not in physics or in nature, is that a problem? AG

Is that an interesting problem? I guess so.

Some theories in mathematics assume an axiom of infinity, like in set theory, analysis, etc.

That has often led to paradoxes, but they have been “solved” by diverse means. So most such theories are considered not being problematic. We can also show that, even restricted on the arithmetical truth (which has no axiom of infinite, as all natural numbers are conceived to be finite), adding an axiom of infinity lead to stronger provability abilities. The set theory ZF proves much more than the arithmetic theories PA, even on just the numbers relations. Yet ZF, and actually all effective theories are limited on a small spectrum of the arithmetical reality. The omega-initial segment of ZF mirrors PA faithfully.

In physics, the universe itself could be infinite, without having any infinite things in it, like the model of Arithmetic (all numbers are finite, and the set of all numbers is just a meta-concept, not representable directly in the theory, but still manageable (you can prove in PA that there is an infinite of prime numbers, by proving

For x (prime(x) -> It exist y (y bigger than x) and prime(y)).

Are there actual infinite object in the universe?

I can prove that if mechanism is false, then there as such object. With Mechanism, the mind is infinite, and physics is somehow the Mind seen from itself internally. That might favours an infinite physical universe. Does our substitution level depend on Planck Constant? Open problem.

With mechanism, the axiom of the infinite is inconsistent on the ontological level, but is a theorem on the phenomenological level. It shortens the proofs, and provide many tools to to handle mathematically the semantic, the notion of limit, many form of approximation, even to learn just about the natural numbers or the combinators.

Mechanism provides a testable account of the mind-body relation, an account which does not assume more than elementary arithmetic, and which doesn’t involve any other ontological commitment. So let us see. The quantum structure, and time, admits a “simple" arithmetical interpretation, but space, dimension, energy remains in the shadow.

Science has not yet decided between Plato's and Aristotle’s conception of reality, and not all people are aware of the hypothetical nature of those options, nor that Digital Mechanism, or even just the Church-Turing thesis, leads us back to Pythagorus and Plato.

The natural numbers realised the infinities without making them into existing things, or beings.

Bruno

--
You received this message because you are subscribed to the Google Groups "Everything List" group.
To unsubscribe from this group and stop receiving emails from it, send an email to everything-li...@googlegroups.com.
To view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/5be11cc0-b02f-48b5-a190-f282fef5a063%40googlegroups.com.

[Philip Thrift]

On Friday, August 30, 2019 at 10:10:22 AM UTC-5, Bruno Marchal wrote:

On 30 Aug 2019, at 04:33, Alan Grayson <agrays...@gmail.com> wrote:

If there are infinities in mathematics, but not in physics or in nature, is that a problem? AG

Is that an interesting problem? I guess so.

Some theories in mathematics assume an axiom of infinity, like in set theory, analysis, etc.

That has often led to paradoxes, but they have been “solved” by diverse means. So most such theories are considered not being problematic. We can also show that, even restricted on the arithmetical truth (which has no axiom of infinite, as all natural numbers are conceived to be finite), adding an axiom of infinity lead to stronger provability abilities. The set theory ZF proves much more than the arithmetic theories PA, even on just the numbers relations. Yet ZF, and actually all effective theories are limited on a small spectrum of the arithmetical reality. The omega-initial segment of ZF mirrors PA faithfully.

In physics, the universe itself could be infinite, without having any infinite things in it, like the model of Arithmetic (all numbers are finite, and the set of all numbers is just a meta-concept, not representable directly in the theory, but still manageable (you can prove in PA that there is an infinite of prime numbers, by proving

For x (prime(x) -> It exist y (y bigger than x) and prime(y)).

Are there actual infinite object in the universe?

I can prove that if mechanism is false, then there as such object. With Mechanism, the mind is infinite, and physics is somehow the Mind seen from itself internally. That might favours an infinite physical universe. Does our substitution level depend on Planck Constant? Open problem.

With mechanism, the axiom of the infinite is inconsistent on the ontological level, but is a theorem on the phenomenological level. It shortens the proofs, and provide many tools to to handle mathematically the semantic, the notion of limit, many form of approximation, even to learn just about the natural numbers or the combinators.

Mechanism provides a testable account of the mind-body relation, an account which does not assume more than elementary arithmetic, and which doesn’t involve any other ontological commitment. So let us see. The quantum structure, and time, admits a “simple" arithmetical interpretation, but space, dimension, energy remains in the shadow.

Science has not yet decided between Plato's and Aristotle’s conception of reality, and not all people are aware of the hypothetical nature of those options, nor that Digital Mechanism, or even just the Church-Turing thesis, leads us back to Pythagorus and Plato.

The natural numbers realised the infinities without making them into existing things, or beings.

Bruno

Also, some may think that because theoretical physics (field theories) is expressed in a language that includes a mathematically continuous (real number) background R^4 of spacetime and the methods of multidimensional calculus (tensor calculus, etc.), that because mathematically infinite-divisibility is present and infinitary definitions (like "limit") are present, that these  "infinities" of the mathematics are real in the actual world.

@philipthrift

 

[Bruno Marchal]

I agree. Infinities are useful does not entails that infinities exist in “Reality”.

That is why I can be more open to your fictionalism for second order arithmetic, analysis, set theory, more than for arithmetic (that would make Mechanism and even just computer science into fiction)..

Bruno

@philipthrift

 

--
You received this message because you are subscribed to the Google Groups "Everything List" group.
To unsubscribe from this group and stop receiving emails from it, send an email to everything-li...@googlegroups.com.

To view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/8728448c-eab1-4e12-86ee-4e3b765b9237%40googlegroups.com.

[Philip Thrift]

cf.  

    

    Joel David Hamkins @JDHamkins https://twitter.com/JDHamkins

     http://jdh.hamkins.org/ 

(I think he has to be a fictionalist.)

@philipthrift