The position of constructive mathematics on the axiom of infinity is outright unsustainable

[Skepdick]

You must assume that such infinitely repeating process is possible and that it will terminate.

So I must delude myself?

I must pretend that if I took a pen to paper I can add infinitely many numbers together?
If that were true why don't you just do it? Why do you keep writing "..."?

How about we replace the axiom of infinity with the axiom of no-self-delusion?

Suddenly the "..." in mathematical notation becomes an honest acknowledgment of ongoing process rather than a hand-wave toward a completed infinity.

Of course, you have to give up the ontological view of Mathematics for a process ontology. But so what?
You lose nothing of mathematical power and you gain something so much richer.

That clearly amounts to assuming that you can traverse the set of the natural numbers from beginning till end, which is exactly what the axiom of infinity proclaims.

I have no problem with the traversing part.
I also have no problem with traversing more; or traversing slower e.g when traversing from 0 to 1 in N there's not much scenery, but in R there is.
I have a problem with coming to an end part.

Seriously, the axiom of infinity is here to stay.

I don't believe you. Now that you have finished defining an infinite object.

Constructive mathematics is nothing but an outrage at time-complexity blindness.

What we have is *ONLY* potential infinities. That's just co-induction and co-recursion + lazy evaluation.
We have co-data types - unfoldable structures rather than completed collections.
We have co-algebras.

You can work with infinite sets that admit algorithmic exploration - where "exhaustive search" means "my process will eventually find any element that exists," not "my process will finish examining all elements."

This gives you the mathematical power you need while avoiding the metaphysical baggage of completed infinities.

https://ieeexplore.ieee.org/document/4276587

The infinity is in the potential of the search process, not in any completed collection being searched.
This much more intellectually and philosophically satisfying than deluding myself about completing infinite tasks.

[Skepdick]

You must assume that such infinitely repeating process is possible and that it will terminate.

In fact, lets assume that! Lets unfold the natural numbers to completion OK?

Just humor me and for every number you unfold toggle a binary switch.
0 -> On
1 -> Off
2 -> On
3 -> Off
...

Now that the infinite process has terminated - tell us if the switch is on; or off.

What's |N| mod 2?

You wanted to know where the contradiction in Classical Mathematics is?
It's right in the axiom asserting the existence of something that cannot exist: a finished infinity.

And now that your on/off switch is in superposition, think about which philosophy of mathematics is actual deserving of the title "Mathematical Realism."

[Eodnhoj7]

You do understand that a purely abstract entity exists only within the observer's head

I disagree. I subscribe to Platonism:

https://plato.stanford.edu/entries/plat ... athematics

Platonism about mathematics (or mathematical platonism) is the metaphysical view that there are abstract mathematical objects whose existence is independent of us and our language, thought, and practices. Just as electrons and planets exist independently of us, so do numbers and sets. And just as statements about electrons and planets are made true or false by the objects with which they are concerned and these objects’ perfectly objective properties, so are statements about numbers and sets. Mathematical truths are therefore discovered, not invented.

as their is no physical medium to share it through the senses?

The interest rate on the 10-year US treasury bill and the exchange rate of JPY/USD are pure abstractions. There is no physical medium to share these things through the senses. Still, they are very, very real. Even by the mainstream media, these things are considered much more relevant than most material objects. They do not just exist within the observer's head.

There is no physical medium to share a tariff through the senses. So, a tariff only exists in Donald Trump's head. Therefore, the Chinese should not waste their time negotiating about mere fictions ... but apparently, they do!

A ten year treasury bill may be dependent upon abstraction but it is a physical entity and not a pure abstraction.

The tariff exists through symbols, writing on paper or digital programs. It also exists through the physically grounded act of communication.

A pure abstraction would have no corresponding physical symbol. If there is a physical aspect to an abstraction than the abstraction is not purely an abstraction.

[godelian]

Constructive mathematics is nothing but an outrage at time-complexity blindness.
What we have is *ONLY* potential infinities. That's just co-induction and co-recursion + lazy evaluation.

Time complexity applies to physical computations, not to abstract ones. In the physical universe, there are only potential infinities. In the Platonic universe, there are actual infinities. I am perfectly fine with actual infinities in the Platonic realm.

[godelian]

Just humor me and for every number you unfold toggle a binary switch.
0 -> On
1 -> Off
2 -> On
3 -> Off
...

Now that the infinite process has terminated - tell us if the switch is on; or off.

That is the same question as: What is sin(α) for α->∞?
Same answer: indeterminate.

[Skepdick]

That is the same question as: What is sin(α) for α->∞?

No, it's not.

Same answer: indeterminate.

Contradiction.

If the process is complete it's either 1 or 0. By LEM.

[Skepdick]

Time complexity applies to physical computations, not to abstract ones.

Yeah, nonsense. It applies to all symbol manipulations.

Lie to me some more that adding 1+1 takes you the same time as 1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1

You are effectively claiming that the input size to the sum() function doesn't affect its runtime. e.g that this runs in O(1) irrespective of inputs.
But that's just false. It's O(n)

Code: Select all

In [1]: sum([1,1])
In [2]: sum([1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
   ...: ,1,1,1])

[godelian]

If the process is complete it's either 1 or 0. By LEM.

Yes, but that does not mean that you know what the answer is. I can run a computer program that flips between 0 and 1 and then stop the program. If I don't print out the last state, I will not know if the program ended with a 0 or a 1.

[godelian]

Time complexity applies to physical computations, not to abstract ones.

Yeah, nonsense. It applies to all symbol manipulations.

Lie to me some more that adding 1+1 takes you the same time as 1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1

On a physical device, it takes time to carry out additions. On an abstract device, it doesn't.

[Skepdick]

Yes, but that does not mean that you know what the answer is. I can run a computer program that flips between 0 and 1 and then stop the program. If I don't print out the last state, I will not know if the program ended with a 0 or a 1.

OK, so you are, in fact claiming that the answer to |N| mod 2 is a Boolean. You just don't know which of the two Booleans?
You are in fact claiming that there will be a last state TO print; and the print out will have a 1; or a 0?

You are, in fact, claiming that somewhere along the process of completing an infinity, a toggle (which is determinate at every step!) lost its determinacy?

[Skepdick]

On a physical device, it takes time to carry out additions. On an abstract device, it doesn't.

So ask your abstract device for instant answers to your math homework.

Can I give you a call so you can instantly factor some 2048-bit numbers for me?

[godelian]

On a physical device, it takes time to carry out additions. On an abstract device, it doesn't.

So ask your abstract device for instant answers to your math homework.

Can I give you a call so you can instantly factor some 2048-bit numbers for me?

ChatGPT: What is the difference between abstract and physical computations?

The distinction between abstract and physical computations centers on what is doing the computing and how:

Abstract Computations

Definition:
Abstract computation refers to idealized, mathematical processes performed by theoretical models of computation.

Characteristics:
Performed by formal systems (e.g., Turing machines, lambda calculus, register machines).
Defined in terms of symbol manipulation rules.
Timeless and spaceless—not constrained by physics.
Used in logic, mathematics, and computer science theory.

Example:
A Turing machine simulating addition of two numbers on an infinite tape.

Physical Computations

Definition:
Physical computation refers to actual, real-world processes performed by physical systems (like computers, brains, or even unconventional systems like DNA molecules).

Characteristics:
Governed by the laws of physics (e.g., thermodynamics, electromagnetism).
Includes concerns like speed, energy, noise, reliability.
Requires hardware or physical media.
Subject to resource limitations (time, memory, power).

Example:
A digital computer calculating π to a million digits using electrical circuits.

[Skepdick]

On a physical device, it takes time to carry out additions. On an abstract device, it doesn't.

So ask your abstract device for instant answers to your math homework.

Can I give you a call so you can instantly factor some 2048-bit numbers for me?

ChatGPT: What is the difference between abstract and physical computations?

The distinction between abstract and physical computations centers on what is doing the computing and how:

blah blah blah

On a physical device, it takes time to carry out additions. On an abstract device, it doesn't.

Dude. I don't care about the lengthy verbiage. I am simply looking at the implications of your words.
Spare me the philosophical smoke and mirrors.

Abstract devices perform zero-latency computations. Correct?
The size of the input does not affect the latency. Correct?

Factoring large numbers is as easy as 1=1.

[godelian]

Dude. I don't care about the lengthy verbiage. I am simply looking at the implications of your words.

ChatGPT is implicitly very Platonic. You can reject Platonism, but then we should simply conclude that this is the core of the disagreement, and that there is nothing we can do about that.

[Skepdick]

Dude. I don't care about the lengthy verbiage. I am simply looking at the implications of your words.

ChatGPT is implicitly very Platonic. You can reject Platonism, but then we should simply conclude that this is the core of the disagreement, and that there is nothing we can do about that.

The core of the disagreement is not philosophical.

The core of the disagreement is your inability to produce instant factorizations to 2048 bit RSA keys.