Theists Equivocating the Empirical with the Transcendental

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bobmax
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Re: Theists Equivocating the Empirical with the Transcendental

Post by bobmax »

godelian wrote: Wed Jun 22, 2022 3:25 am After familiarizing myself quite a bit with the matter, I feel that I cannot and do not want to argue against this body of mathematical knowledge. Seriously, I do not have the credibility to attack this.

Therefore, I will not seek to go in opposition against the very foundations of the mathematical field.

Good luck arguing against standard modern mathematics!
It is possible to show how, following Cantor's same diagonal argument, the same system of natural numbers is uncountable...

Once all the infinite natural numbers have been listed, a matrix is ​​constructed by planning as many non-significant zeros as needed to the left of each number.

Cantor places them on the right of the infinite real numbers from zero to one, we put them on the left of the natural numbers.
(These zeros are not significant in both cases.

We apply the Cantor diagonal argument... and the natural numbers are not themselves countable!
More absurd than that.
godelian
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Re: Theists Equivocating the Empirical with the Transcendental

Post by godelian »

Immanuel Can wrote: Wed Jun 22, 2022 5:40 am
godelian wrote: Wed Jun 22, 2022 3:43 am
Immanuel Can wrote: Tue Jun 21, 2022 5:38 pm But the question is, can one have an actual infinite.
Modern mathematics is not about the physical universe.
That's only partly true. Some mathematics is purely formal.

Mathematical concepts are just that: concepts. However, they have an adjectival relationship to nouns...they just indicate how many of something are in view, not what those things are. Because mathematics can contribute adjectival information about nouns, they can speak adjectivally about the real world. They can be used to condition or describe empirical facts.

"Two sheep" is a real descriptor of an empirical reality. Mathematics doesn't provide the noun, just the quantity. The empirical supplies the fact that they are sheep.

Or, to take a different example, if one starts from Earth and moves outward in a linear direction, one can keep going for infinity, presumably, because the universe is expanding.

The question of infinity, as it pertains to the empirical world, is "Can there be an infinite regress in a chain of causes?" And mathematics shows us that the answer is "No."
Other fields, that are empirical, use mathematics as a language and as a tool to maintain consistency in what they say.
For example, science does that. Engineering does that too. Quite a few other fields do that too.

It is important to consider, that a scientific formula about the physical universe is an expression that belongs to science and not to mathematics. If the semantics of a logic sentence are not about abstract, Platonic objects, then it is not mathematics, but something else.

Example:
- 5 is not divisible by 2 ==> mathematics
- 5 sheep cannot fit into a standard van built for only 4 ruminants ==> not mathematics (only some of the use of language is governed by mathematics here)
godelian
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Re: Theists Equivocating the Empirical with the Transcendental

Post by godelian »

Veritas Aequitas wrote: Wed Jun 22, 2022 5:42 am Strawmaning, I did not attack modern mathematics as useless.
I did not say that you argued that modern mathematics would be useless. (That is actually the real straw man here.)
I argued that your attack on religion eventually always degenerates into an attack on mathematical Platonism, which is fundamental to modern mathematics.
godelian
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Re: Theists Equivocating the Empirical with the Transcendental

Post by godelian »

bobmax wrote: Wed Jun 22, 2022 6:25 am It is possible to show how, following Cantor's same diagonal argument, the same system of natural numbers is uncountable...
The existence of uncountable (non-standard) natural numbers as universes generated by Peano Arithmetic is usually argued on grounds of the Lowenheim-Skolem theorem ("upwards" and "downwards"):
Victoria Gitman on "nonstandard models of arithmetic" wrote:
An introduction to nonstandard models of arithmetic

It already follows from the Lowenheim-Skolem theorems that there are uncountable models of the Peano axioms. Yes, there are uncountable structures in which induction holds! Let's call the natural numbers the standard model of the Peano axioms and all other models nonstandard. An easy application of the compactness theorem shows that there are countable nonstandard models of the Peano axioms, or indeed of any collection of true arithmetic statements. Order-wise, these models look like the natural numbers followed by densely many copies of the integers: N followed by Q-many copies of Z (see the slides for explanation).
bobmax wrote: Wed Jun 22, 2022 6:25 am We apply the Cantor diagonal argument... and the natural numbers are not themselves countable!
More absurd than that.
There is a MathExchange question about exactly this:
Question:

Why doesn't Cantor's diagonal argument also apply to natural numbers?
...
If we could, then the diagonal argument would imply that there is a natural number not in the natural numbers, which is a contradiction.

Answer:

The reason is simply that natural numbers have a finite representation. (In set theory, Each one is a finite number of successions from 1, where the successor of n is n+1.) Your representation scheme essentially respects this, since for any natural number (which will be on the list), after some finite number of digits it becomes all zeros. Your diagonal element will either be one of these, and so on the list, or a sequence of ones and zeros which never 'zeros out'. This string is NOT a natural number; it corresponds to nothing in your representation scheme. The reason the diagonal argument works for real numbers is that they do not have a finite representation.
So, there are indeed uncountable models for arithmetic theory, filled with nonstandard natural numbers, on grounds of Lowenheim-Skolem. However, you cannot legitimately argue on grounds of Cantor's diagonal argument that the standard natural numbers would be uncountable.

If you really want to consider something absurd about the natural numbers:
Victoria Gitman on "nonstandard models of arithmetic" wrote: In particular, a nonstandard model of arithmetic can have indiscernible numbers that share all the same properties.
bobmax
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Re: Theists Equivocating the Empirical with the Transcendental

Post by bobmax »

godelian wrote: Wed Jun 22, 2022 7:30 am There is a MathExchange question about exactly this:
Question:

Why doesn't Cantor's diagonal argument also apply to natural numbers?
...
If we could, then the diagonal argument would imply that there is a natural number not in the natural numbers, which is a contradiction.

Answer:

The reason is simply that natural numbers have a finite representation. (In set theory, Each one is a finite number of successions from 1, where the successor of n is n+1.) Your representation scheme essentially respects this, since for any natural number (which will be on the list), after some finite number of digits it becomes all zeros. Your diagonal element will either be one of these, and so on the list, or a sequence of ones and zeros which never 'zeros out'. This string is NOT a natural number; it corresponds to nothing in your representation scheme. The reason the diagonal argument works for real numbers is that they do not have a finite representation.
So, there are indeed uncountable models for arithmetic theory, filled with nonstandard natural numbers, on grounds of Lowenheim-Skolem. However, you cannot legitimately argue on grounds of Cantor's diagonal argument that the standard natural numbers would be uncountable.
How can there be a finite representation of infinite numbers?

The representation is said to be finite because it assumes that it can go to infinity by treating it as a thing.

So you go back to the starting point.

The infinite is arbitrarily actualized.

It's just a ploy to avoid being chased out of Cantor's mathematical paradise.
Iwannaplato
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Re: Theists Equivocating the Empirical with the Transcendental

Post by Iwannaplato »

godelian wrote: Wed Jun 22, 2022 7:07 am
Veritas Aequitas wrote: Wed Jun 22, 2022 5:42 am Strawmaning, I did not attack modern mathematics as useless.
I did not say that you argued that modern mathematics would be useless. (That is actually the real straw man here.)
I argued that your attack on religion eventually always degenerates into an attack on mathematical Platonism, which is fundamental to modern mathematics.
And just to restress a point I made earlier. He is conflating
can be traced back to [empirical processes]
with
is fundamentally empirical.

That is confused. It is an equivocation on 'fundamentally'. Sure, in time, but not ontologically.

Of course advanced primates are going to do (start doing) math with processes like counting things.

But this is NOT an argument against mathematical realism, or platonic ontologies of math.

It's like saying something like

physics was Newtonian in the past, so fundamentally physics is Newtonian. So, there are no quantum effects. The universe is classical. You cannot have things like entanglement or wave particle dualism.

The history of a process of learning has NOTHING to do with the implications of later knowledge. and the ontologies discovered, say, therein.

It's an implicit argument in his posts. IOW he does not carry out all the steps. He thinks that by saying that it can be traced back to the empirical, in the process of learning all math is actually empirical and there are no platonic aspects to it. But he doesn't write the full argument. And if he did he might notice the problem.
Last edited by Iwannaplato on Wed Jun 22, 2022 8:54 am, edited 2 times in total.
Skepdick
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Re: Theists Equivocating the Empirical with the Transcendental

Post by Skepdick »

bobmax wrote: Wed Jun 22, 2022 6:04 am It looks like an induction but it isn't.

Because the law that derives from an induction is already present in the cases that are observed.

I always see white swans so I induce that all swans must be white.
Induction is not logically compelling.
Well, obviously - how can you induce anything about the unobserved?!?
bobmax wrote: Wed Jun 22, 2022 6:04 am In fact it can always happen to run into a black swan, and my law lapses.
Precisely. So until you observe an infimum of he Natural numbers, until you observe an N that doesn't have an N+1, until you falsify the law - it holds!
bobmax wrote: Wed Jun 22, 2022 6:04 am While here we only observe that if I add 1 I get a new number.
I am observing a logical rule.
And as such it admits no possibility of being proven wrong.
And if you ever encounter a number to which you can't add 1 then the rule is falsified.

Is there a number to which you can't add 1?
bobmax wrote: Wed Jun 22, 2022 6:04 am That the numbers are unlimited is therefore a further step, that is, a deduction.
That is, I deduce that I will never be able to reach the end of this process.
It's not deduction. The entity that can add 1 to get a new number is always YOU.

If you stop adding at 10 it's not because you can't add 1 to get to 11. It's because you chose to stop at 10.
The point being - you can always chose to go one step further.
And one more.
And one more.
...
Skepdick
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Re: Theists Equivocating the Empirical with the Transcendental

Post by Skepdick »

bobmax wrote: Wed Jun 22, 2022 8:19 am How can there be a finite representation of infinite numbers?
Trivially. In programming languages.

From https://en.wikipedia.org/wiki/Kolmogorov_complexity
In algorithmic information theory (a subfield of computer science and mathematics), the Kolmogorov complexity of an object, such as a piece of text, is the length of a shortest computer program (in a predetermined programming language) that produces the object as output.
The shortest computer program in the language of Haskell which represents the set of Natural numbers is thus...

Code: Select all

[0..]
And then you can then take as many numbers from the set as you need.

Code: Select all

Prelude> take 5 [0..]
[0,1,2,3,4]
Prelude> take 27 [0..]
[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26]
Skepdick
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Re: Theists Equivocating the Empirical with the Transcendental

Post by Skepdick »

Iwannaplato wrote: Wed Jun 22, 2022 8:42 am And just to restress a point I made earlier. He is conflating
can be traced back to [empirical processes]
with
is fundamentally empirical.

That is confused. It is an equivocation on 'fundamentally'. Sure, in time, but not ontologically.

Of course advanced primates are going to do (start doing) math with processes like counting things.

But this is NOT an argument against mathematical realism, or platonic ontologies of math.

It's like saying something like

physics was Newtonian in the past, so fundamentally physics is Newtonian. So, there are no quantum effects. The universe is classical. You cannot have things like entanglement or wave particle dualism.

The history of a process of learning has NOTHING to do with the implications of later knowledge. and the ontologies discovered, say, therein.

It's an implicit argument in his posts. IOW he does not carry out all the steps. He thinks that by saying that it can be traced back to the empirical, in the process of learning all math is actually empirical and there are no platonic aspects to it. But he doesn't write the full argument. And if he did he might notice the problem.
A curious observation - in the argument above the concept/notion of a "process" is being given an ontological treatment - it's a first class citizen. A necessary existent for the argument to work.

SO.

What's a process?

https://en.wikipedia.org/wiki/Process_(computing)
https://en.wikipedia.org/wiki/Process_calculus

In as far as you are all confused and in seek of foundations... look no further than computer science.
Skepdick
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Re: Theists Equivocating the Empirical with the Transcendental

Post by Skepdick »

godelian wrote: Wed Jun 22, 2022 3:38 am Reverse mathematics requires foundationalism, if only, because it seeks to establish new foundations.
Those are not the goals of reverse mathematics though?

And can you really call it "foundationalism" if every theory I develop (every computer program I write) is "founded" upon different premises?
godelian wrote: Wed Jun 22, 2022 3:38 am Why would it do that, if it didn't believe in the need for foundations?
Because it believes in the need of sufficiency.

Personally, the only "foundation" I need to develop any Mathematical theory is to reject every single axiom of logic/Mathematics. Especially the non-contradiction axiom.

Unencumbered by the limits to non-contradiction I am free to use the untyped lambda calculus; or fixed point combinators.
You know - all the unsound, contradiction-producing tools Mathematicians hate.
godelian wrote: Wed Jun 22, 2022 3:38 am Hence, reverse mathematics is the epitome of foundationalism!
It sure looks like anti-foundationalism from where I am looking.
godelian wrote: Wed Jun 22, 2022 3:38 am If you don't like football, then play tennis or so, instead.
Precisely. And I don't like foundationalism. Which is why I am playing anti-foundationalism.

There are infinitely-many incommensurable "foundations" (arbitrary choices) to Mathematics.
bobmax
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Re: Theists Equivocating the Empirical with the Transcendental

Post by bobmax »

Skepdick wrote: Wed Jun 22, 2022 8:45 am And if you ever encounter a number to which you can't add 1 then the rule is falsified.

Is there a number to which you can't add 1?
bobmax wrote: Wed Jun 22, 2022 6:04 am That the numbers are unlimited is therefore a further step, that is, a deduction.
That is, I deduce that I will never be able to reach the end of this process.
It's not deduction. The entity that can add 1 to get a new number is always YOU.
Do you see the contradiction in what you say?

It is the essence of a natural number that 1 can be added to it.

There is no need for any proof, there can be no doubt.
Because it is a tautology.

I can't run into a natural number to which I can't add 1.
Because it wouldn't be a natural number.

The computer can never prove anything.
I've worked a lifetime with computers, already with the early Intel 8085s.
The computer, like our human logic, cannot go beyond its finitude.
Skepdick
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Re: Theists Equivocating the Empirical with the Transcendental

Post by Skepdick »

bobmax wrote: Wed Jun 22, 2022 10:47 am Do you see the contradiction in what you say?
No, I don't. Because there is no contradiction in what I am saying.

bobmax wrote: Wed Jun 22, 2022 10:47 am It is the essence of a natural number that 1 can be added to it.
Not if they are finite! Then it would be the essence of all natural numbers EXCEPT the largest one.

bobmax wrote: Wed Jun 22, 2022 10:47 am There is no need for any proof, there can be no doubt.
Because it is a tautology.
The word "tautology" means "true in ALL models."

You are the one claiming that there is a model of the natural numbers which is NOT infinite.
e.g YOU are the one saying that there is a model of the natural numbers in which there is an N for which N+1 is not allowed.
bobmax wrote: Wed Jun 22, 2022 10:47 am I can't run into a natural number to which I can't add 1.
Because it wouldn't be a natural number.
It would be a natural number. A special natural number - the largest one!

If you are now insisting that there doesn't exist a model of the natural numbers in which there is a largest natural number then you believe that the natural numbers are infinite!

How is it possible that you believe these two things at the same time?

1. You believe that what I am saying is true in all models.
2. You believe that there is at least one model in which what I am saying is not true.

bobmax wrote: Wed Jun 22, 2022 10:47 am The computer can never prove anything.
I've worked a lifetime with computers, already with the early Intel 8085s.
The computer, like our human logic, cannot go beyond its finitude.
Then you have absolutely no idea what a "proof" is.

https://en.wikipedia.org/wiki/Curry%E2% ... espondence
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Immanuel Can
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Re: Theists Equivocating the Empirical with the Transcendental

Post by Immanuel Can »

godelian wrote: Wed Jun 22, 2022 7:02 am It is important to consider, that a scientific formula about the physical universe is an expression that belongs to science and not to mathematics.
That's arbitrary, really. Anything that's a "formula" is mathematical in structure, if not in referent. Inversely, pure mathematics that has no reference to reality is an imaginary exercise conducted within an artificial, closed system of symbols. So the matter is not so tidy, I think. And historically, we know that the truth is that mathematical adjectives are derived from empirical properties...men needed symbols for counting sheep.

Science and maths have a cooperative relation. And truths evident in mathematics are not going to fail to be reflected in empirical realities.

For example, if mathematics says a particular operation cannot be rendered in a mathematically-accurate way, it's not as if that reality is going to exist in the scientific realm.

So if mathematics shows us, as it does, that an infinite regress of prerequisites cannot exist, then it's not as if the universe can possibly still be the product of an infinitely receding chain of causes. The maths simply expose the empirical truth of that.
godelian
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Re: Theists Equivocating the Empirical with the Transcendental

Post by godelian »

Skepdick wrote: Wed Jun 22, 2022 9:27 am Unencumbered by the limits to non-contradiction I am free to use the untyped lambda calculus; or fixed point combinators.
You know - all the unsound, contradiction-producing tools Mathematicians hate.
The problem with the untyped lambda calculus revolves around Haskell Curry's paradox.
Wikipedia on "Curry's paradox" wrote: Curry's paradox is a paradox in which an arbitrary claim F is proved from the mere existence of a sentence C that says of itself "If C, then F", requiring only a few apparently innocuous logical deduction rules. Since F is arbitrary, any logic having these rules allows one to prove everything. The paradox may be expressed in natural language and in various logics, including certain forms of set theory, lambda calculus, and combinatory logic.The paradox is named after the logician Haskell Curry. It has also been called Löb's paradox after Martin Hugo Löb,[1] due to its relationship to Löb's theorem.
The paradox is closely connected to the problem of unrestricted set comprehension in set theory:
Even if the underlying mathematical logic does not admit any self-referential sentences, certain forms of naive set theory are still vulnerable to Curry's paradox. In set theories that allow unrestricted comprehension, we can nevertheless prove any logical statement Y ...
Standard Set Theory (ZFC) hedges against that problem by introducing axioms that outlaw unrestricted set comprehension:
Step 4 is the only step invalid in a consistent set theory. In Zermelo–Fraenkel set theory, an extra hypothesis stating X is a set would be required, which is not provable in ZF or in its extension ZFC (with the axiom of choice).

Therefore, in a consistent set theory, the set { x ∣ x ∈ x → Y } does not exist for false Y. This can be seen as a variant on Russell's paradox, but is not identical. Some proposals for set theory have attempted to deal with Russell's paradox not by restricting the rule of comprehension, but by restricting the rules of logic so that it tolerates the contradictory nature of the set of all sets that are not members of themselves. The existence of proofs like the one above shows that such a task is not so simple, because at least one of the deduction rules used in the proof above must be omitted or restricted.

Curry's paradox can be formulated in any language supporting basic logic operations that also allows a self-recursive function to be constructed as an expression. Two mechanisms that support the construction of the paradox are self-reference (the ability to refer to "this sentence" from within a sentence) and unrestricted comprehension in naive set theory. Natural languages nearly always contain many features that could be used to construct the paradox, as do many other languages. Usually the addition of meta programming capabilities to a language will add the features needed.
Allowing this in the untyped lambda calculus has the same effect as allowing unrestricted set comprehension in ZFC. It makes the theory inconsistent:
In the 1930s, Curry's paradox and the related Kleene–Rosser paradox played a major role in showing that formal logic systems based on self-recursive expressions are inconsistent. These include some versions of lambda calculus and combinatory logic.

Lambda calculus may be considered as part of mathematics if only lambda abstractions that represent a single solution to an equation are allowed. Other lambda abstractions are incorrect in mathematics.

Curry's paradox and other paradoxes arise in Lambda calculus because of the inconsistency of Lambda calculus considered as a deductive system.

Lambda calculus is a consistent theory in its own domain. However, it is not consistent to add the lambda abstraction definition to general mathematics.

Resolution in set theory

In Zermelo–Fraenkel set theory (ZFC), the axiom of unrestricted comprehension is replaced with a group of axioms that allow construction of sets. So Curry's paradox cannot be stated in ZFC. ZFC evolved in response to Russell's paradox.
The untyped lambda calculus is inconsistent (Curry's paradox) in the same way as naive set theory is (Russell's paradox). Naive set theory is simply standard set theory minus the axioms that specifically rein in the problem:

- axiom schema of restricted comprehension
- Axiom of regularity (also introduced to avoid Mirimanoff's non-well-founded sets)

In principle, set theory has the same problem as the lambda calculus, but in set theory the problem can be reined in by adding axioms that deal with it. However, even a set theory improved with axiomatic restrictions cannot add axioms that will successfully defeat Gödel's incompleteness theorems.
Skepdick
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Re: Theists Equivocating the Empirical with the Transcendental

Post by Skepdick »

godelian wrote: Wed Jun 22, 2022 2:18 pm The problem with the untyped lambda calculus revolves around Haskell Curry's paradox.
I am familiar with Curry's paradox. What I don't understand is why you think this paradox, or any paradox is a "problem".

It's just a theorem. It's true that allowing conditionals such as "if A, then B" is sufficient to prove any B.

Or if you want me to say this in a Reverse Mathematical setting. What are the sufficient conditions for proving any B?
Answer: Conditionals in the form of "if A, then B"
godelian wrote: Wed Jun 22, 2022 2:18 pm The paradox is closely connected to the problem of unrestricted set comprehension in set theory
Nor do I understand why you think unrestricted comprehension is a "problem".
godelian wrote: Wed Jun 22, 2022 2:18 pm Standard Set Theory (ZFC) hedges against that problem by introducing axioms that outlaw unrestricted set comprehension
I didn't realise there were legal and moral authorities in Mathematics. So much so that you could "outlaw" stuff?
godelian wrote: Wed Jun 22, 2022 2:18 pm Allowing this in the untyped lambda calculus has the same effect as allowing unrestricted set comprehension in ZFC. It makes the theory inconsistent
Sure. And? Are you willing to give up unrestricted comprehension for the sake of consistency? It's just an engineering trade-off - a design choice.

Consistency, inconsistency, para-consistency, completeness, incompleteness, decidability, semi-decidability etc. are just semantic properties of the theory itself. Surely people can design their theories to exhibit whatever semantic properties their hearts desire?

You can design a consistent-but-incomplete theory about the numbers.
You can design a complete-but-inconsistent theory about the numbers.
You just can't design a consistent AND complete theory about the numbers.

What I see is a choice, but I don't see a problem?
godelian wrote: Wed Jun 22, 2022 2:18 pm In principle, set theory has the same problem as the lambda calculus, but in set theory the problem can be reined in by adding axioms that deal with it. However, even a set theory improved with axiomatic restrictions cannot add axioms that will successfully defeat Gödel's incompleteness theorems.
Perhaps it's best you define what you mean by "problem", because it sure seems to me that you think a whole bunch of semantic properties/features of formal systems are "problematic". You seem to think those features are bugs - so much so that you are willing to limit the expressive power of your system just to avoid them.

Could it be that you carry some sort of an axiomatic bias?
Could it be that you hold an axiom so dear that you wouldn't consider giving it up?
Could it be that you have a sacred cow that you would never consider slaughtering?
Could you be religious?
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