I would not equate mathematical structure with logical structure, nor even, what seems at least prima facie more plausible, arithmetic with logic. I do not think one can be reduced to the other.

What's that in English. (Serious question.)

Really? It's clear that the natural numbers can be constructed from the logical idea of a set. 0 = {}, 1 = {{}}, 2 = {{}, {{}}}, etc. Then arithmetic can be stated as a list of logical axioms.

To my (very limited) knowledge, everything can be constructed from there. The reals via equivalence classes of Cauchy sequences, so forth.

And Godel's proof of the Incompleteness Theorem relied on converting logical systems to numbers, IIRC.

I thought logic == math was pretty much a given, to be honest.

This looks a lot like cogito ergo sum to me.

This can't be right.

That is just semantic. Any theory A can be countered with an opposing theory B ("theory A is not true"). Proving one wrong means proving the other one true.

Descartes said cogito ergo sum - I think, therefore I am.

The effect is of course the same, but my version has the notable advantage of not introducing a false sense of agency. (That is, the immediate perception that a "thought" is distinct from a "thinker.")

Ofcourse things can be proven, but the nature of the proof will not be absolute. If you need absolute knowledge to accept a proof then...er...you don't do much.

That being said the collective works of philosophy through the ages deals mainly with this question.

In math given the basic axioms theorems can be proven. The axioms must be accepted without proof or on faith.

In any closed system with rules, axiomatic or not, absolute proofs are possible. So that means formal logic and mathematics are candidates for absolute certainty.

Asking whether the axioms are provable is a category mistake.

I don't entirely agree. We can (and, in practice, do) use a limited notion of proof based on common sense. It goes like this:
* If a claim cannot be falsified by any analogous precedent, then that claim is proven true. *

For example:
* When I step out my front door in a few minutes to go practice piano, I will not fall into the Grand Canyon. *

There is no precedent for buildings spontaneously relocating to the edge of the Grand Canyon. So in the limit of common sense, my claim is proven true.

Sorry for spamming your thread, Our House.

Seriously? This is the law of the contrapositive: http://en.wikipedia.org/wiki/Contrapositive

We have to rely on our senses to know that the sentence "This sentence exists" exists. Descartes does not think that he must rely on his senses to know that "I am thinking, so I must exist" is true.

There's a lot to be said about criticisms of the cogito, along the lines of those introduced by the likes of Lichtenburg and later Russell, viz., that it is illicit to infer a thinker, but I think it is hard to deny that thoughts logically imply a thinker, and I tend to side with those who argue that such criticisms rely on a much-too-crude empiricism.

I do think that incompleteness makes it difficult if not impossible to reduce mathematics to logic, at least not without serious modifications to the project as it was originally conceived by Russell and Whitehead. A brief discussion:

http://www.askphilosophers.org/question/2244

If you can define 'proof' without invoking this experience of 'common sense certainty' to get your essential meaning, then I will withdraw my objection.

Dr Johnson kicking a stone in the street in response to Berkeley's theory that nothing exists outside the mind: "I refute it thus".

Sorry for the disappearance...I had to step out for a bit.

There's a bunch of fancy talk and fancy characters in this thread that I don't really understand, so I'll try to explain my OP in simpleton terms if you guys don't mind.

I'm working with the assumption that we exist and everything/everyone we see around us also exists. No brains in a vat, no complex philosophy, none of that stuff. No circular definitions (e.g. "A bicycle has two wheels") and no statements about statements (e.g. "Can you falsify falsifiability?"). On the surface it seems like a lot of conditions, but I'd like to keep the frame of reference as "realistic" as possible so that we can all relate to it what we're trying to prove as fact.

Now, from what I understand, mathematical concepts can be proven. They can be worked out to be shown 100% true. This doesn't apply to scientific theories, or anything for that matter that depends on the outcome of predictions. That's basically what I'm asking. Can we ever be 100% certain that something will or will not happen, or is everything in life based on confidence intervals?

Hmm, it seems arbitrary to draw a line anywhere and say, "Up to this point, my perception depends on my senses; any deeper, and the senses are unnecessary."

All perception consists of analogy to or memory of sense experience. (In Descartes' case, obviously we could point to his past sensory use of the symbols 'I', 'think', etc.; and all the sensory contexts where those symbols had appeared.)

I agree, empiricism got off to a terrible start because its earliest advocates lacked nuance. However, I think this more a critique of those thinkers rather than the methodology.

Correct me if I'm wrong, but this appears to be the money quote here:

I'm not convinced this is much of an objection. Perhaps I'm missing something huge, but it seems bizarre to say that a symbolic isomorphism could be "covering up" a disjoint ontology.

Not much of a disjunction, in my mind!

To add to my last post...

We can show something to be 100% false, but can we ever (outside of math) show something to be TRUE?

Truth by itself is not a big deal. I could come up with any crazy original definition of a new math structure X and prove certain things about it and they would be 100% true. That's e.g. what a lot of math PhD's look like. But it is not the point. It's how applicable the definition is, either in math or science (preferably both), and how rich the theory it generates that matters.

What you seem to be asking is if math theories are 100% applicable in science.

im sure there is some postulate about pointless threads that we are proving right now.

Sure. I know that. I want Philo to elaborate on whatever point he was alluding to.

Your version has a related problem in that a sentence can only exist if there's someone to interpret it as a sentence.

Mathematics needs "things". That's one reason for the linked thread.

How many "things" are there in the universe?

http://forumserver.twoplustwo.com/47...iverse-344017/

Yeah, I misread that post into thinking it wasn't what it actually was. Never mind.

Proving true or false is just a matter of semantics.

K, sorry, didn't mean to state the obvious. No offence, but you asked for plain English, and this was not what you asked for in plain English.

That may be your opinion, but is nowhere implied in the statement: "This sentence exists."

On the other hand, Descartes explicitly said, "I think...", etc.

Do you think you need to use your senses in order to have the thought "I am thinking" or to know that you are thinking?

I meant only that Descartes thought that he could be certain that he was thinking as much as he could be certain of anything. In fact, Descartes thought the cogito was more certain than that 2+2=4.

He did not, however, think of the cogito as a kind of proof, or indeed as involving any sort of inference at all. He thought that it was "clear and distinct" and did not think that there was any inference from "I think" to "I exist".