If you find the word "rationalistic" insulting — Gregory

It was not 'rationalistic'; it was 'over rationalistic'. And it's not that I find it so insulting, but that it is ad hominem.

Anyhow are you saying they can find a limited number of theorems that they know for sure is provable from the ground up? — Gregory

Godel discusses formal theories. What does "provable from the ground up" mean in regards to Godel's incompleteness theorem?

I merely repeated my previous post on Gödel. — Gregory

"

Gödel already proved that mathematics is either wrong or that there are infinite things that can't be proven. — Gregory

and

mathematics was inconsistent (aka wrong) or incomplete — Gregory

are critically different.

It sounds like you make logic or math a religion — Gregory

Of course I don't. I listed a good number of the major features of religion. I don't relate to mathematics with any of those features.

I said "seem" which means "appears to others' perception" — Gregory

Oh please, a minor pedantic shift in the way one casts an ad hominem doesn't erase it's affect as an hominem.

1931 paper he said he set out to prove that mathematics was inconsistent (aka wrong) or incomplete — Gregory

And that doesn't deserve the mangled version you posted earlier.

Where can the line be drawn with which to distinguish the provable from the unprovable? — Gregory

It's not clear what "line to be drawn" means there. And every formula is provable in some system or another.

What ad hominem? — Gregory

Your claim that I seem overly rationalistic, obviously.

I feel like you ask for proof for the obvious — Gregory

Ah, argument by "it's obvious".

it's correct — Gregory

Where did Godel say it?

You responded to the wrong person — Gregory

No, I didn't.

wrote like a Logicist — Gregory

That's idiotic. Logicism is the view that mathematics can be derived from logical axioms alone. I've never even flirted with logicism.

It's what you feel, not what you know. — jgill

If I understand you correctly, sure, one can feel that one is being religious. I am not disputing that he is accurately describing the way he feels or even views what he does.

Just to be clear, that ad hominem doesn't refute anything I've said. Second, I have all kinds of facets, not just reason, but in discussing mathematics, I prefer to be lucid.

Gödel already proved that mathematics is either wrong or that there are infinite things that can't be proven. — Gregory

That is one of the worst encapsulations of Godel I've ever read.

The problem is that they can never find where the line is between what is provable and what is not — Gregory

Another botched attempt.

We can list the essential attributes of religion, and see which of those are attributes of mathematics. Attenuated, glib, "kinda sorta" analogies like "axioms are a fiats" or "infinity is mystic" don't count.

Mathematicians do sometimes say cheeky things like that. But mathematics doesn't have religious practices, rites, rituals, creeds, obligations of obedience, eschatology, myths, scripture, moral strictures, determination of truth by fiat and divine revelation, etc.

Mathematics is the opposite of religion.

Darn, I was hoping to hear about a work in mathematics that is from a dead universe perspective and moves in infinite circles.

And now I'm also hoping to hear of a study that shows that scientists have an especially high incidence of mental illness.

Then I was wondering what you had in mind when you mentioned Godel's theorems, but the only thing so far you've mentioned is his and Cantor's breakdowns. If you are suggesting that Godel and Cantor suffered mental illness due to their work in mathematics, then that would require evidence. Also, I hope you are not suggesting that Godel's and Cantor's personal problems discredit the mathematics.

I explained to you already why bijection (paring) is an inadequate representation of counting — Metaphysician Undercover

And at every juncture I pointed out where you are wrong or confused.

I don't know what a "set" is, you haven't defined it. — Metaphysician Undercover

x is a set iff (x is the empty class or (x is a non-empty class and there is a y such x is a member of y)).

Or, the sets are objects that satisfy the set theory axioms.

Or, the sets are the objects that the quantifier ranges over.

But you seemed to be using it as if it meant the result of the count — Metaphysician Undercover

No, I did not. I have always been completely clear that the bijection represents the count, not the result. You are terribly terribly confused.

Isn't it the case, that the mathematical representation of a count, is the number, which is the result of the count? — Metaphysician Undercover

No! And I've told you this already. What is wrong with you? The mathematical representation of the count is a representation of the count, not of the result. The representation of the count is the bijection. The result of the count is a number.

Would you give an example of a work in mathematics that is from a dead universe perspective and moves in infinite circles?

Mathematics is usually practiced as a Platonic type of religious practice. — Gregory

No, it's not.

I find no definite sense from your use of all that undefined terminology and assumptions. And I surmise that continuing to ask you will lead to only more.

But I am intrigued what you have in mind regarding Godel's theorems.

Logicism is often thought to have failed because it was not found how to derive mathematics without non-logical axioms.

That does not refute what I said about the rigor of mathematics.

I know what ordinals and density are. I just want to know what you mean by 'infinite density' in mathematics.

And I know that the philosophy of mathematics is a field of study.

It's just about numbers in general vs sets which have infinite density — Gregory

So anything anyone says about numbers in general is mathematics? And if your philosophizing about that mathematics is purely philosophical then that mathematics is not all true?

And what do you mean by 'infinite density'? Is that a mathematical notion of yours or purely philosophical?

The natural numbers?

They're not a doubling sequence. They're a successorship sequence.

And what theory of the natural numbers? There are many.

then mathematics is not one hundred percent truth — Gregory

What do refer to when you say 'mathematics'?

Sure, in a purely philosophical context, you can come up with all kinds of stuff.

Mathematics is rigorous by effectivized formal languages, recursive axiom sets, and recursive inference rules, and explicit statements of algorithms for checking for well-formedness, axiomhood, and proof.

Mathematics has both finite and infinite sets.

Your notion of a mathematics with only infinite sets is purely fanciful without at least an outline of the primitives and axiomatic notions.

If mathematics talks about an order which is not temporally, nor spatially grounded, then I think such a mathematics would be nonsensical. — Metaphysician Undercover

The mathematics of ordering and ordinals may be applied to study of space and time, but the mathematics itself doesn't mention space and time.

do you not agree that the "thing" is an object? — Metaphysician Undercover

I agree that things are objects. In my previous post, I answered essentially the same question, when I said 'Yes'.

Do you agree that the thing is a "unity"? — Metaphysician Undercover

I already shared my thoughts about 'unity' earlier in this thread.

(3)

If you simply say "1,2,3,4,5" , you might say "I am counting", but it's not a true count — Metaphysician Undercover

That would be another sense of the English word 'count', and it may be represented mathematically as

<1 2 3 4 5>

But it was not the sense in your bookshelf example, which may be represented mathematicaly as the bijection I mentioned.

Unless the person knows what the symbols mean they are not really counting to five — Metaphysician Undercover

I don't have an opinion about that.

When I gave a mathematical representation of a count.
— TonesInDeepFreeze

Please, do not jump ahead like that. — Metaphysician Undercover

What? I'm not jumping ahead. I'm referring back. You asked me where the notion of 'set' came from in this discussion, so I told you.

There is no need to represent (2), the result of the act of counting, as a "set" — Metaphysician Undercover

You are critically confused on the very point here, and one that previously you even said you understood. That point is that the result is different from the count. I didn't represent the result as a set*. I explicity said (several times) that the result is a number. Meanwhile I represented the count (not the result) as a bijection, which is a certain kind of set.

(* Putting aside the technical sense of all objects as sets in formal set theory.)

For all x, y, z, if x=y and y=z, then x=z.

It's famous that monadic languages lack the expressiveness of dyadic languages, and that monadic logic is weaker than predicate logic with dyadic predicates.

So I responded to your challenge. Howzabout you respond to mine from previous posts?:

https://thephilosophyforum.com/discussion/comment/533894

If ExP is true, then that requires a task, call it TaskE.
If ~ExP is true, then that requires a task, call it TaskN.
TaskE and TaskN are different.
— TonesInDeepFreeze
Sorry, you're just repeating yourself. — InPitzotl

It's fair for you to have quoted me that way, since I did post it. But, just for the record, around the same time, I edited my post to not include that, as it's wrong, and I misspoke earlier when I said they are different.

1. All As are Bs
2. No As are Bs
3. Some As are Bs
4. Some As are not Bs

I'm told that every proposition can be rephrased as one of the above. — TheMadFool

You were told wrong.

"'the task P if the Goldbach conjecture is true' is a different task than 'the task P if the Goldbach conjecture is false'"? — InPitzotl

Let's go back the general question about ExP.

I'm not couching this as "The task for proving ExP when ExP is true is different from the task for proving ExP when ExP is false."

What I am saying is this: Proving ExP is "easy" only if ExP is true.

I wouldn't say ExP is easy. Because then someone may say, "It's not easy if it's false, because its's impossible, which is the ultimate not easy."

So I include the antecedent "If ExP is true".

And I'm not saying that is interesting. It's just necessary to be correct.

And to make meaningful comparison between proving ExP and ~ExP, we need to consider each when it is true.

I'll do one of Earl Hines's "Blues In Thirds".

the latter is dependent on spatial-temporal relations — Metaphysician Undercover

For physical world matters. However, in the mathematics itself, ordinals don't refer to space and time.

If you count "1", then it is implied that there is one thing (an object) counted. Do you, or do you not agree with this? — Metaphysician Undercover

Agree.

If you are counting books, then aren't books objects? — Metaphysician Undercover

Yes.

it is necessary that an object has been counted? Therefore an object is implied by any count of 1? — Metaphysician Undercover

I just told you that I don't use the 'implied' that way.

In your post you said, "it is implied that there is one thing". And that is how I use 'imply' too. I use 'imply' to say 'It is implied that [fill in statement here].

Then you said, "an object is implied".

I don't use 'implied' to say '[fill in noun phrase here] is implied'.

When did a "set" enter the picture? — Metaphysician Undercover

When I gave a mathematical representation of a count.

Let's look at Turing machine framework (I think I have this right):

Suppose P is a computable property of natural numbers. (Analogously, for purpose of this discussion, we suppose "this is a dog and it's black" is a definite enough statement that we can definitively declare when we find a black dog.)

Ask the machine for 'yes' or 'no' to "Is ExP true"?

If ExP is true, then the machine will answer 'yes'.

If ExP is false, then the machine might not halt.

If ~ExP is true (i.e. ExP is false), then the machine might not half.

If ~ExP is false (i.e. ExP is true), then the machine will answer 'yes'.

Let me rephrase. They are the same task. But if ExP is true, then the task is sure to end, while if ~ExP is true, then its end is indeterminate.

This holds for this framework we're talking about - empirical search, one-by-one in an indeterminately large domain.

And there's an analogy to it in mathematics [I'm simplifying somewhat]:

Let P be a computable property of natural numbers.

If ExP is true, then we are ensured that in finite time we will find an x such that we prove Px is true, thus proving ExP.

But even if ~ExP is true, then we are not ensured that we will ever prove ~ExP (it might be the case that at all points in time, indefinitely, we don't know whether it's provable).

They cannot both be true.

These principles have been offered, where the scope is not determined:

AxP is falsifiable but not verifiable
ExP is verifiable but not falsifiable

I think that is reasonable, if we take 'falsifiable' and 'verifiable' in a sense of 'definitively'. But if we admit degrees of falsification and degrees of verification, then perhaps we would adjust the above principles proportionately. But for the moment I'll take the notions in the sense of 'definitively'.

Also, reiterating what has already been mentioned:

~AxP is equivalent with Ex~P, so it is verifiable but falsifiable.
~ExP is equivalent with Ax~P, so it is falsifiable but not verifiable.

The relevant comparison is between proving ExP when it is true vs. proving ~ExP when it is true. (For a falsehood, not only is it difficult to prove, but it is impossible to prove.)

Also, if discovery of proof proceeds by one-by-one examination of things, then yes, if ExP is true, then the sequence of proving by one-by-one examination for ExP is finite, while, if ~ExP is true, then the sequence of proving by one-by-one examination for ~ExP is indeterminate. And that holds with the example of "There is a black dog" vs. "There is not a black dog". They are not the same task.

So it has been claimed that this difference entails that the first burden is on ExP. It seems there might be something to that, but it is not self-evident and it requires support.

But we also want to consider cases where the scope is determinate and a context in which verification and falsification are not definitive but refer to degrees of verification and degrees of falsification. In either of those two frameworks, we can easily see that sometimes proving ExP when it is true is not "easier" than proving ~ExP when it is true.

/

Regarding whether there exits an omnipotent, omnipresent, omnibenevolent being, I'm not saying anything new here, but for me, the question requires specifying what would constitute empirical proof. If it's not an empirical matter, and unless the existence statement is shown to be a logical truth, then it seems it's a metaphysical or theological concern for which the notion of proof in the same sense of proving "there exists a black dog" doesn't even apply.

When I say 'P is implied', then P is a statement, not an object.

So I don't say

'War And Peace' is implied.

But I do say

That 'War And Peace' is on the bookshelf is implied.

This is just a matter of being very careful in usage that may be critical in discussions about mathematics.

Regarding this example of counting, I take it as a given assumption that

'War And Peace' is on the bookshelf and 'Portnoy's Complaint' is on the bookshelf.

I am not deriving ''War And Peace' is on the bookshelf and 'Portnoy's Complaint' is on the bookshelf' as implied by anything other than the initial assumption of the example.

And, of course, I am not showing an example of a non-empty count on the empty set. It is a given assumption of the example that:

the set of books on shelf = {'War And Peace' 'Portnoy's Complaint'}

/

Stipulation does not make truth. — Metaphysician Undercover

I knew you would respond in a way that would evince that you don't understand the concept of definition.

First, there is no general definition of number in mathematics.
— fishfry

That's because numbers are not objects — Metaphysician Undercover

No, your belief that numbers are not objects is not the reason that mathematics doesn't provide a definition of 'is a number'.

Yes, I was not qualifying your remark regarding the implication of the counterfactual.

Of course, the gambler's fallacy cannot be mathematically true. But, of course, if the gambler's fallacy turned out to be empirically true, and the casinos did not adjust their payout ratios, then the casinos would lose (and even if people didn't even play with the gambler's fallacy in mind).