Theists Equivocating the Empirical with the Transcendental

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Skepdick
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Re: Theists Equivocating the Empirical with the Transcendental

Post by Skepdick »

bobmax wrote: Tue Jun 21, 2022 5:15 pm Regardless of whether abstract or not, the set of natural numbers can never be infinite.
It is in the very nature of the infinite not to be there.

Infinity is an open idea, it cannot be taken and used as a thing.
In an act of rebelion I shall demonstrate otherwise.

Here's a piece of code written in Python.
It implements what's called a continuation

In so far as the function integers() does anything - it implements the infinite set of natural numbers. Which ones? ALL of them.
Only, instead of returning all of the integers all at once, it returns them one by one (on-demand, as-needed)

Code: Select all

In [1]: def integers():
   ...:     n = 0
   ...:     while True:
   ...:         yield n
   ...:         n = n + 1
   ...:

In [2]: n = integers()
In [3]: next(n)
Out[3]: 0
In [4]: next(n)
Out[4]: 1
In [5]: next(n)
Out[5]: 2
To convince yourself of this fact you are welcome to keep calling the function next() and convince yourself that in so far as you can think of any number you can always call next() once more. So you will never run out of numbers.

Said diffently: Infinity is always one more than what you need. Infinity can simply be thought of "always having enough".

And if you want to convince yourself that the set is indeed infinite then you can trivially construct an infinite loop which calls next() forever...

Code: Select all

In [6]: while True:
   ...:     print(next(n))
   ...:
   3
   4
   5
   6
   7
   8
   9
   <If you have any patience this will go on forever>
bobmax
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Re: Theists Equivocating the Empirical with the Transcendental

Post by bobmax »

Skepdick wrote: Tue Jun 21, 2022 10:10 pm
bobmax wrote: Tue Jun 21, 2022 5:15 pm Regardless of whether abstract or not, the set of natural numbers can never be infinite.
It is in the very nature of the infinite not to be there.

Infinity is an open idea, it cannot be taken and used as a thing.
In an act of rebelion I shall demonstrate otherwise.

Here's a piece of code written in Python.
It implements what's called a continuation

In so far as the function integers() does anything - it implements the infinite set of natural numbers. Which ones? ALL of them.
Only, instead of returning all of the integers all at once, it returns them one by one (on-demand, as-needed)

Code: Select all

In [1]: def integers():
   ...:     n = 0
   ...:     while True:
   ...:         yield n
   ...:         n = n + 1
   ...:

In [2]: n = integers()
In [3]: next(n)
Out[3]: 0
In [4]: next(n)
Out[4]: 1
In [5]: next(n)
Out[5]: 2
To convince yourself of this fact you are welcome to keep calling the function next() and convince yourself that in so far as you can think of any number you can always call next() once more. So you will never run out of numbers.

Said diffently: Infinity is always one more than what you need. Infinity can simply be thought of "always having enough".

And if you want to convince yourself that the set is indeed infinite then you can trivially construct an infinite loop which calls next() forever...

Code: Select all

In [6]: while True:
   ...:     print(next(n))
   ...:
   3
   4
   5
   6
   7
   8
   9
   <If you have any patience this will go on forever>
I don't think it proves infinity.
Rather, it shows the unlimited.

That is, it shows that there is no limit.

To prove infinity it would take an infinite amount of time.
Then the infinite is mistakenly actualized, but it is only a misunderstanding, that is, the infinite is arbitrarily reduced to a thing.

Infinity is impossible to show, because it doesn't exist.
However, we can realize the unlimited.
And this happens when we fail to get to where the finite ends.

For example, the universe is unlimited, for the simple reason that it is finite and it ends in the nothing.
Skepdick
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Re: Theists Equivocating the Empirical with the Transcendental

Post by Skepdick »

bobmax wrote: Tue Jun 21, 2022 10:38 pm I don't think it proves infinity.
Rather, it shows the unlimited.

That is, it shows that there is no limit.
Potato, potatoh.

Different words - same idea.
bobmax wrote: Tue Jun 21, 2022 10:38 pm To prove infinity it would take an infinite amount of time.
This requirement is incoherent. A proof is a program. To "prove" something requires the proof (program) to terminate.
And if a program terminates it necessarily means it has only run for a finite amount of time - it has only performed a finite number of operations.

So you seem to have it backwards - any terminating proof is a finite proof. Thus what is sufficient to demonstrate infinity is any non-terminating proof (program).

Such as this infinite loop which will run for an infinite amount of time.

Code: Select all

while True:
   pass
if you insist that I have not met the burden of proof, then I insist you prove that the program (above) terminates.

If it terminates (there's a limit to its runtime) - it's finite.
If it doesn't terminate (there is no limit to its runtime) - it's infinite.
bobmax wrote: Tue Jun 21, 2022 10:38 pm
Then the infinite is mistakenly actualized, but it is only a misunderstanding, that is, the infinite is arbitrarily reduced to a thing.
Where is the "mistake" in the program I've given you?

The infinite has been reduced to a finite representation - sure, but those two lines of Python amount to an infinite loop.

bobmax wrote: Tue Jun 21, 2022 10:38 pm Infinity is impossible to show, because it doesn't exist.
Then you need to explain what it is that you are looking at if not an infinte loop?

And if you insist that it's not infinite - please predict its termination time/date.
bobmax wrote: Tue Jun 21, 2022 10:38 pm For example, the universe is unlimited, for the simple reason that it is finite and it ends in the nothing.
If the universe is unlimited then there is no limit to where it ends (nowhere).

Infinity and unlimitedness are the same idea.

The moment you find the limit of the universe though, it becomes finite.
bobmax
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Re: Theists Equivocating the Empirical with the Transcendental

Post by bobmax »

Skepdick,

With that function you will never be able to use infinity.
Infinity is always and only in potentiality, it is never actual.

Cantor actualizes it, erroneously.

Instead, you can verify that there are no limits.
But this is not because you actually go to infinity, you verify it by deducing it.

But a deduction absolutely does not imply that you can have access to infinity.
It is just an illusion.

A very dangerous illusion.
Skepdick
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Re: Theists Equivocating the Empirical with the Transcendental

Post by Skepdick »

bobmax wrote: Tue Jun 21, 2022 11:08 pm Skepdick,

With that function you will never be able to use infinity.
With that function I will be able to use as much of infinity as I need.

I am reaonably certain I'll never run out, but I don't rule out the possibility I may be wrong. Time will tell.
bobmax wrote: Tue Jun 21, 2022 11:08 pm Infinity is always and only in potentiality, it is never actual.
In so far as it never runs out; and there's always more of it than I need - it's actual.

If I need numbers (unique identifiers) I will be able to obtain as many of those as I need!
bobmax wrote: Tue Jun 21, 2022 11:08 pm Instead, you can verify that there are no limits.
Why do I have to repeat myself? Unlimited and infinite are synonymous.

If you find some upper limit to the naturals then yeah - you are right. But they are unlimited.
bobmax wrote: Tue Jun 21, 2022 11:08 pm But this is not because you actually go to infinity, you verify it by deducing it.
But a deduction absolutely does not imply that you can have access to infinity.
Not at all! I am veryfying it by induction, not deduction.

Is there always one more number? Yes. Every N has an N+1.

Since I can't think of an N which has no N+1 then whatever I have access to is infinite.
bobmax wrote: Tue Jun 21, 2022 11:08 pm It is just an illusion.

A very dangerous illusion.
Or a very useful instrument.

But here's the deal. If the numbers are not infinite then they must have an infimum. What is it?
Skepdick
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Re: Theists Equivocating the Empirical with the Transcendental

Post by Skepdick »

godelian wrote: Tue Jun 21, 2022 3:30 pm This is a first blow to the nominalist-empiricist view on mathematics, as modern mathematics does indeed implement Aristotle's foundationalism completely and throughout. Modern mathematics reasons from unproven First Principles, i.e. axioms,
That's a very constrained view of Mathematics you have there - there is more to it than axiomatics.

Specifically Reverse Mathematics starts with the theorems first and then figures out which axioms are necessary/sufficient.
godelian
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Re: Theists Equivocating the Empirical with the Transcendental

Post by godelian »

bobmax wrote: Tue Jun 21, 2022 5:15 pm
godelian wrote: Tue Jun 21, 2022 4:13 pm Now, natural numbers are not physical objects. Therefore, there is no compelling reason to assume that it would take time, effort, or energy to traverse them from beginning till end. Let's assume that we can fully traverse them. Now let's name the cardinality of this set: countable infinity ("aleph-0").
It is this assumption that is wrong.
Because it is always possible to add another number.

Regardless of whether abstract or not, the set of natural numbers can never be infinite.
It is in the very nature of the infinite not to be there.

Infinity is an open idea, it cannot be taken and used as a thing.

It doesn't matter if abstract or not, infinity cannot be elaborated precisely because of its intrinsic impregnability.

Infinity cannot even be really thought of.
One always thinks only of the finite.

And it is always the finite which, by denying it, generates the infinite.
But only as a negation of the finite.
Not because you can really think of infinity...

Infinity is just pure negation.
And every negation draws all its raison d'etre in what it denies.
I understand that you do not want to accept the axiom of infinity.

https://en.m.wikipedia.org/wiki/Axiom_of_infinity

This axiom is included in Zermelo-Fraenckel (ZF or ZFC) which has been the standard for set theory for over a century now:
Wikipedia on "Axiom of infinity" wrote: In axiomatic set theory and the branches of mathematics and philosophy that use it, the axiom of infinity is one of the axioms of Zermelo–Fraenkel set theory. It guarantees the existence of at least one infinite set, namely a set containing the natural numbers. It was first published by Ernst Zermelo as part of his set theory in 1908.[1]
The reason why it has to be an axiom, is because it is provably independent from the other axioms of set theory:
Wikipedia on "the independence of the infinity axiom" wrote: The axiom of infinity cannot be proved from the other axioms of ZFC if they are consistent. (To see why, note that ZFC & Con(ZFC – Infinity) and use Gödel's Second incompleteness theorem.)

The negation of the axiom of infinity cannot be derived from the rest of the axioms of ZFC, if they are consistent. (This is tantamount to saying that ZFC is consistent, if the other axioms are consistent.) We believe this, but cannot prove it (if it is true).
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!
godelian
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Re: Theists Equivocating the Empirical with the Transcendental

Post by godelian »

Skepdick wrote: Tue Jun 21, 2022 11:43 pm
godelian wrote: Tue Jun 21, 2022 3:30 pm This is a first blow to the nominalist-empiricist view on mathematics, as modern mathematics does indeed implement Aristotle's foundationalism completely and throughout. Modern mathematics reasons from unproven First Principles, i.e. axioms,
That's a very constrained view of Mathematics you have there - there is more to it than axiomatics.

Specifically Reverse Mathematics starts with the theorems first and then figures out which axioms are necessary/sufficient.
Reverse mathematics requires foundationalism, if only, because it seeks to establish new foundations. Why would it do that, if it didn't believe in the need for foundations?

Hence, reverse mathematics is the epitome of foundationalism!

Furthermore, in this matter, I fully subscribe to the formalist view that the axioms are merely symbol strings that can legitimately be replaced by other, even arbitrary symbol strings.

The symbols that constitute the axioms are not "about" anything at all. New axioms are symbols that are still not "about" anything at all. Therefore, any set of old axioms is not "better" than the new ones.

Peano Arithmetic Theory (PA) and Zermelo-Fraenckel Set Theory (ZFC) are very popular and even dominant but not "better" in any sense.

These are merely axiomatic environments in which a lot of mathematicians like to work. We are talking about just some kind of preference here. I have never said that other axiomatizations would be "wrong" or so.

If you don't like football, then play tennis or so, instead.
godelian
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Re: Theists Equivocating the Empirical with the Transcendental

Post by godelian »

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.

Therefore, since the question pertains the physical universe, it will never be answered by modern mathematics.

Seriously, "actual infinity" is not our problem, simply because the physical universe is not our problem.

If you want to deal with the physical universe, then you will need to consult a field like physics instead. It is them who are interested in that kind of questions, not us.
godelian
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Re: Theists Equivocating the Empirical with the Transcendental

Post by godelian »

Skepdick wrote: Tue Jun 21, 2022 9:52 pm Which Universe in the Multiverse of Mathematics is the Platonic Universe?
Arithmetic Theory generates one "standard" universe, i.e. the natural numbers, i.e. its so-called "intended interpretation".

Arithmetic Theory also generates an infinite number of nonstandard universes with nonstandard numbers.

So, on the whole, Arithmetic Theory generates a multiverse and not a simple universe.

With Arithmetic Theory (PA) and Set Theory (ZF-inf) being bi-interpretable, they generate exactly the same standard universe (of standard natural numbers).

Concerning the bi-interpretability of their nonstandard universes, and therefore of the complete multiverse, I am not sure, because the publication "On Interpretations of arithmetic and set theory" by Kaye and Wong, does not mention anything about that.

I personally suspect that the complete multiverse of natural numbers and natural sets may also be bi-interpretable, but we will need to wait for a publication that formally proves that.

Therefore, the abstract, Platonic world of mathematics is a multiverse, with at its core the universe of standard natural numbers (or equivalent standard natural sets).

This multiverse also has an infinite number of additional nonstandard universes that contain nonstandard numbers (or equivalent nonstandard sets).

So, mathematics is about an abstract, Platonic multiverse.
Veritas Aequitas
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Re: Theists Equivocating the Empirical with the Transcendental

Post by Veritas Aequitas »

godelian wrote: Tue Jun 21, 2022 10:05 am
Veritas Aequitas wrote: Tue Jun 21, 2022 7:08 am
Kant in CPR wrote: The Concept of 12 is by no means already Thought in merely Thinking this union of 7 and 5; and I may analyse my Concept of such a possible sum as long as I please, still I shall never find the 12 in it.
We have to go outside these Concepts, and call in the aid of the Intuition which corresponds to one of them,
our five fingers, for instance,
From the above, it implies there are no platonic '5' '7' or '12' existing independently by themselves to be discovered by humans.
If you want to understand the nature of the standard universe of the natural numbers, read up on Peano Arithmetic Theory (PA) and the Lowenheim-Skolem theorem which will give you an introduction to modern model theory and which clarifies that there exist nonstandard universes that interpret PA, and therefore, that PA generates a multiverse:
Wikipedia on "nonstandard models of arithmetic" wrote: In mathematical logic, a non-standard model of arithmetic is a model of (first-order) Peano arithmetic that contains non-standard numbers. The term standard model of arithmetic refers to the standard natural numbers 0, 1, 2, …. The elements of any model of Peano arithmetic are linearly ordered and possess an initial segment isomorphic to the standard natural numbers. A non-standard model is one that has additional elements outside this initial segment. The construction of such models is due to Thoralf Skolem (1934).

There are several methods that can be used to prove the existence of non-standard models of arithmetic.

- From the compactness theorem.
- From the incompleteness theorems.
- From an ultraproduct.
Therefore, the idea that the only interpretation of arithmetic theory would be the natural numbers, is utterly simplistic and even ignorant. Furthermore, it is not possible to discover the existence of non-standard numbers by counting your fingers. Again, there are serious limits to aboriginal empiricism. Therefore, we cannot use that approach in modern mathematics. Aboriginal empiricism must be utterly rejected.

Furthermore, there is no way that you can understand that there exist logic sentences that are true in the natural numbers but not provable from PA (Gödel's incompleteness theorem), if you stick to your 18th century view on arithmetic. Kant's simplistic analysis of the natural numbers has been completely superseded in the two centuries that followed the publication of CPR.
Veritas Aequitas wrote: Tue Jun 21, 2022 7:08 am
Wikipedia on "mathematical Platonism" wrote: A major question considered in mathematical Platonism is: Precisely where and how do the mathematical entities exist, and how do we know about them? Is there a world, completely separate from our physical one, that is occupied by the mathematical entities? How can we gain access to this separate world and discover truths about the entities? One proposed answer is the Ultimate Ensemble, a theory that postulates that all structures that exist mathematically also exist physically in their own universe.
Platonic ideas, form and universals are highly contentious and easily demonstrated to be unreal.
Kant had demonstrated Plato ended up in la la land; [mine]
Thus the idea that mathematical entities exist by themselves out there is absurd.
Where you insist mathematical entities [platonic] exist as real by themselves independent of human conditions, that is delusional.
Btw, which "shoulders of giants" [philosophers] are you standing on to support your views above?
You seem to dislike mathematical Platonism. However, that does not matter, because that will not change anything to the fact that it is the dominant ontology for mathematics amongst mathematicians.
plato.stanford.edu on "Mathematical Platonism" wrote: Platonism in the Philosophy of Mathematics

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.

The most important argument for the existence of abstract mathematical objects derives from Gottlob Frege and goes as follows (Frege 1953). The language of mathematics purports to refer to and quantify over abstract mathematical objects. And a great number of mathematical theorems are true. But a sentence cannot be true unless its sub-expressions succeed in doing what they purport to do. So there exist abstract mathematical objects that these expressions refer to and quantify over.

1. What is Mathematical Platonism?

Mathematical platonism can be defined as the conjunction of the following three theses:

Existence.
There are mathematical objects.

Abstractness.
Mathematical objects are abstract.

Independence.
Mathematical objects are independent of intelligent agents and their language, thought, and practices.

1.2 The philosophical significance of mathematical platonism

Mathematical platonism has considerable philosophical significance. If the view is true, it will put great pressure on the physicalist idea that reality is exhausted by the physical. For platonism entails that reality extends far beyond the physical world and includes objects which aren’t part of the causal and spatiotemporal order studied by the physical sciences.[1] Mathematical platonism, if true, will also put great pressure on many naturalistic theories of knowledge. For there is little doubt that we possess mathematical knowledge. The truth of mathematical platonism would therefore establish that we have knowledge of abstract (and thus causally inefficacious) objects. This would be an important discovery, which many naturalistic theories of knowledge would struggle to accommodate.

2.4 The notion of ontological commitment

Versions of the Fregean argument are sometimes stated in terms of the notion of ontological commitment. Assume we operate with the standard Quinean criterion of ontological commitment:

Quine’s Criterion.
A first-order sentence (or collection of such sentences) is ontologically committed to such objects as must be assumed to be in the range of the variables for the sentence (or collection of sentences) to be true.

Then it follows from Classical Semantics that many sentences of mathematics are ontologically committed to mathematical objects.
So, according to your own remarks, you know absolutely nothing about mathematical Platonism, while you use the fact that you are completely ignorant of mathematical Platonism as the justification for why it would be wrong. So, no, your views do not not justify that mathematical Platonism would be wrong. Your ignorance on the matter only justifies the claim that you know nothing about it.
Your mind and views are on a one-track-path to nowhere and la la land.

Note we are doing 'philosophy' here not Mathematics per se.
So what is of concern is the Philosophy of Mathematics.

My basic principle re philosophy is,
All Philosophies are Reduced to Realism vs Anti-Realism [Idealism]
viewtopic.php?f=5&t=28643

Whilst you are banking on Platonic Mathematics, it is merely a type of Mathematical Realism.
  • Mathematical realism, like realism-in-general, holds that mathematical entities exist independently of the human mind. Thus, humans do not invent mathematics, but rather discover it, and any other intelligent beings in the universe would presumably do the same.
    https://en.wikipedia.org/wiki/Philosoph ... al_realism

OTOH, my view of mathematics is that of Mathematical Anti-Realism, e.g.
  • In mathematics, intuitionism is a program of methodological reform whose motto is that "there are no non-experienced mathematical truths" (L. E. J. Brouwer).
    From this springboard, intuitionists seek to reconstruct what they consider to be the corrigible portion of mathematics in accordance with Kantian concepts of being, becoming, intuition, and knowledge. Brouwer, the founder of the movement, held that mathematical objects arise from the a priori forms of the volitions that inform the perception of empirical objects.
    https://en.wikipedia.org/wiki/Philosoph ... tuitionism
As I had qualified, "empirical" in his case is linked to the a priori, not merely a posteriori.

So in summary [philosophically],

No matter how you try to argue for Platonic Mathematics, it is merely a form of Mathematical Realism which is fundamentally 'Realism' [Metaphysical / Philosophical Realism] grounded on Pure Reason which ultimately is illusory.

OTOH, I am countering your Mathematical Realism with the basic of Metaphysical Anti-Realism [Kantian] which is realistic and grounded on the a priori [empirical].

I am not saying Platonic Mathematics is useless rather its grounding is illusory.

It is on this same illusory platonic grounds that theists conjure their illusory God and reifying such an illusion as real to the extent that such a god would command theists to kill non-theists to the possibility of exterminating the human race [an ultimate concern of Philosophy-proper].

You OTOH is a one-track-path to nowhere and la la land.
godelian
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Re: Theists Equivocating the Empirical with the Transcendental

Post by godelian »

Veritas Aequitas wrote: Wed Jun 22, 2022 5:03 am Pure Reason which ultimately is illusory.
I have always known, and always argued, that the atheist attack on religion would inevitably degenerate into an attack on Pure Reason.
Veritas Aequitas wrote: Wed Jun 22, 2022 5:03 am It is on this same illusory platonic grounds that theists conjure their illusory God and reifying such an illusion as real ...
I have always known, and always argued, that the atheist attack on religion would inevitably degenerate into an attack on modern mathematics.

I rest my case.
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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 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."
Veritas Aequitas
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Re: Theists Equivocating the Empirical with the Transcendental

Post by Veritas Aequitas »

godelian wrote: Wed Jun 22, 2022 5:23 am
Veritas Aequitas wrote: Wed Jun 22, 2022 5:03 am Pure Reason which ultimately is illusory.
I have always known, and always argued, that the atheist attack on religion would inevitably degenerate into an attack on Pure Reason.
Veritas Aequitas wrote: Wed Jun 22, 2022 5:03 am It is on this same illusory platonic grounds that theists conjure their illusory God and reifying such an illusion as real ...
I have always known, and always argued, that the atheist attack on religion would inevitably degenerate into an attack on modern mathematics.

I rest my case.
Strawmaning, I did not attack modern mathematics as useless. My point is whatever the mathematics, it can be traced fundamentally to the empirical a priori on the basis of Metaphysical Anti-Realism [Kantian].

It was you who went off tangent from the OP with mathematical realism,
viewtopic.php?p=578437#p578437
bobmax
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Re: Theists Equivocating the Empirical with the Transcendental

Post by bobmax »

Skepdick wrote: Tue Jun 21, 2022 11:23 pm
bobmax wrote: Tue Jun 21, 2022 11:08 pm But this is not because you actually go to infinity, you verify it by deducing it.
But a deduction absolutely does not imply that you can have access to infinity.
Not at all! I am veryfying it by induction, not deduction.
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.
In fact it can always happen to run into a black swan, and my law lapses.

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.

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.

But not that there is then a system of infinite natural numbers that I can work out as I please, as Cantor does!
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