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?

Those two sentences together comprehensively state Gödel system without using equations

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.

Well let's put the rationalist thing aside. How do you know numbers have truth value? Do you base mathematics on philosophy or on itself?

I think you are being too literal. — jgill

Too literal about the word 'religious'? It is used as cudgel. So it is good, as a starting point at least, to point out that it is not literally true.

Those two sentences together comprehensively state Gödel system without using equations — Gregory

The second one is reasonable. The first one is a mess.

How do you know numbers have truth value? — Gregory

Whoever said numbers have truth value(s)?

You two are going at it, hot an heavy. I'll steal away.

1 plus 1 equals two only if 1 and 2 exist and can exist. So things boil down to our world view at the end of the day

1 plus 1 equals two only if 1 and 2 exist and can exist. So things boil down to our world view at the end of the day — Gregory

I asked whoever said that numbers have truth value(s)?

Or is it your own claim that numbers have truth value(s)?

All mathematicians says that numbers exist in some sense, which is the same as saying they have truth value. The struggle over the foundations of math is a search for ultimate certainty, much like Descartes's journey in 1641-42. Searching for logical certainty can be a mask for too much doubt

All mathematicians says that numbers exist in some sense, which is the same as saying they have truth value. — Gregory

No, what have truth values are statements.

What have a truth values are the statements "There exists a natural number that is the successor of 0" and "There exists a natural number that is the successor of 1".

The numbers themselves don't have truth values.

This is not a mere pedantic distinction. It is a distinction needed so that the discussion about this subject is coherent.

Searching for logical certainty — Gregory

The theorems are certain or uncertain exactly to the extent that the axioms are certain or uncertain.

It is freely granted that the common axioms are non-logical. That is why it is really stupid for you to say that I'm a logicist (or however exactly you said it).

However, if any non-logical mathematical judgements are certain, then they are those of finitistic combinatorial mathematics. We may be skeptical of finitistic combinatorial mathematics, but then we might as well be skeptical of everything mathematical.

Words and thoughts about numbers are just like any sentence in language and they depend on how much the person values them. Consider that divination is the same as reading philosophy. They are a reading of something through a perceptual experience of our being of-the-world. The only issue is how much knowledge is gained and what is useful for life

Let me add another example I thought of from another thread. Maybe numbers are like energy and they become particles so to speak when they are replaced with infinities. Perhaps this is the next level up and when I said "moving in infinite circles" I meant always trying to prove things in a system where you don't know if what your trying to prove can even be proven

when I said "moving in infinite circles" I meant always trying to prove things in a system where you don't know if what your trying to prove can even be proven — Gregory

For a consistent formal (I always mean 'consistent formal' in this context) system S, there is an infinite enumeration of the proofs. So it is linear, not circular.


In a idealized context without a finite upper limit of time to prove (such as with Turing machines), if S is incomplete, then for a sentence P, at any point in the enumeration, there are these possibilities (proof relative to S):

(1) P been proven.

(2) ~P has been proven.

(3) P is provable, and we will eventually find a proof.

(4) ~P is provable and we will eventually find a proof.

(5) P is not provable and ~P is not provable, and we will never find a proof of P and we will never find a proof of ~P ("eternally floundering to know" whether P is provable and "eternally floundering to know" whether ~P is provable).

(6) In a metatheory for S, we prove "P is provable or ~P is provable".

(7) In a metatheory for S, we prove that P is provable though we don't know a proof itself.

(8) In a metatheory for S, we prove that ~P is provable though we don't know a proof itself.

(9) In a metatheory for S, we prove that P is not provable. (e.g., Cohen proof that AC is not provable from ZF and that CH is not provable from ZFC)

(10) In a metatheory for S, we prove that ~P is not provable. (e.g., Godel proof that ~AC is not provable from ZF and that ~CH is not provable from ZFC)

(11) In a metatheory for S, we prove that P is not provable and that ~P is not provable. (e.g., the conjuction of Godel and Cohen)


For mortal beings, or assuming that in some finite time there won't' be any conscious beings, "eventually" does not hold.

But mathematics itself is very clear about the limitations mentioned in (1)-(11) and makes that lack of conclusiveness itself a subject of rigorous study. This is yet another respect in which mathematics is diametrically different from religion.

I think there are infinite things we can prove with math and infinite things we cant. But math starts with assumptions and that was my point. There are philosophical ways of understanding reality that excludes mathematics and all the assumptions connected to it that sprang from the school of Pythagoras. I'm not saying do away with mathematics. I was offering a way for people see reality that is based more on classical Romantic thought and dialectic than anything traditionally mathematical

I think there are infinite things we can prove with math and infinite things we cant. — Gregory

For any given consistent system S, there are infinitely many theorems and infinitely many non-theorems.

For any given consistent system S, there are infinitely many theorems and infinitely many non-theorems. — TonesInDeepFreeze

Yes. I was just offering a philosophical look at numbers

"Hegel rejected the Newtonian conception of absolute space, arguing that the infinity of space is ideal, but, in disagreement with Kant, Hegel held that this did not require us to accept pure intuitions as uninformed by conceptuality, and therefore did not require us to accept Kant's unfortunate doctrine of the transcendental ideality of space and the dualistic distinction between things-in-themselves and appearances... It is definitely not a matter of whether nature is extended matter and spirit non-extended mental substance. Or, as we have already put it, the distinction between nature and spirit is itself a spiritual". Terry Pinkard

If you read this carefully, you will find a new type of thinking that is different from the tradition of Kant and his predecessors. It's all about logic, infinities, and philosophy