[questor]

In science, we look for truth. this is frequently difficult to determine and faulty conception can ensue as a result. my question is: can we depend upon math as an immutable truth? are the mathematical predictions of theories such as GTR or quantum mechanics always true, or are there discrepancies?
do we have all the math we will ever need to explain everything, or will perhaps some new ways of computation someday arise? there are many things not yet explained or discovered that have been predicted, what do we need that we don't have? will we find that consciousness, instinct and thought will someday be explained by formulae, or will some things never be
predicted by math?

[CraigD]

Originally Posted by questor

… can we depend upon math as an immutable truth? …

Before considering this question, I think it’s important to agree on a precise definition of “Math”. Math can refer to a branch of academia (containing as many specialized sub-branches as “Science”), the consensus views of the mathematical community, a dogmatic educational curriculum, etc. In the context of this thread, I believe Math refers to what one might call “the essence” of Math, that which distinguishes it from other subjects. In my opinion, this essence is captured by the idea of formal systems, as described in such works as Hofstader’s “Godel, Escher, Bach”: that mathematical truth (theorems) can be, and can only be, generated by processes that are independent of human (or any other sentient) influences – that is, that theorems must be generated by algorithms.

Using this definition, we can “depend” on Math to be reproducible, and for the algorithms constituting up a formal system to produce the same result regardless of when or by whom they are done.

In practice, doing Math to this ideal degree of rigor is very difficult, so, just as in less formal disciplines, much of what we accept and reasonably call “Math” lacks even this guarantee. In most cases, when one concludes a mathematical proof with the traditional “Quod Erat Demonstrandum” (QED), one actually means “I’m pretty sure, given enough time, I or someone smarter and/or better trained could represent this as a formal system that could be algorithmically proven true with absolute certainty”.

At the turn of the 20th century, most mathematicians, although they wouldn’t have expressed it in quite these terms, understood Math to be, ideally and practically, what I’ve stated above. What’s more, they believed that all mathematical truth could, by a sufficiently smart person (or, in the extreme, perhaps, by a sufficiently omniscient deity), be reduced to a formal system. It’s nearly impossible, I think, to overstate, then, the significance of Godel’s 1st incompleteness” theorem, (ca. 1930) which proved (to quote the linked article): “For any consistent formal theory including basic arithmetical truths, it is possible to construct an arithmetical statement that is true but not included in the theory. That is, any consistent theory of a certain expressive strength is incomplete.” This introduces a troubling limitation on what we can “depend” on Math to do. We can continue to trust its reproducibility, but we can no longer be sure that a particular formal system will be able to do what we want it to do.

This is not to say that formalism has failed us utterly – in a seemingly paradoxical way, formalism is able to make consistent and complete statements about the limitations of formal systems. The dream of generations of mathematicians of a “universal formalism”, as summarized by such ideas as ”Hilbert’s program”, is not to be – or at least will not much resemble what they imagined.

do we have all the math we will ever need to explain everything, or will perhaps some new ways of computation someday arise? there are many things not yet explained or discovered that have been predicted, what do we need that we don't have?

New ways of computation arise practically every day. A direct consequence of Godel’s theorem is that a “final formal system” of Math – a collection of theorems that can explain, without generating contradictions, anything that can be explained by any formal system, is impossible. Math must constantly grow and adapt.

will we find that consciousness, instinct and thought will someday be explained by formulae, or will some things never be predicted by math?

My personal opinion is yes, that all of these qualities will be formally explained, and soon – by 2050. The social significance of this is moot – even now, a majority of people in all world cultures believe and act on beliefs that are demonstrably counter to evidence and rational thought. A detailed understanding of human consciousness will, I believe, likely be useful to and believed only by a small community of specialists, and considered outright lies and sophistry by many people outside of this community.

… are the mathematical predictions of theories such as GTR or quantum mechanics always true, or are there discrepancies?

Certainly these predictions have discrepancies, both well known ones, and ones yet to be discovered and studied. Among the well-known ones:

• Relativity is “classical” - deterministic and, in mathematical terms, continuous - putting it at odds with Quantum Mechanics (although much work has been done to make QM consistent with Relativity)
• The Standard Model of Particle Physics has not been successfully extended to predict the effect of gravity

to mention just a few.

[HydrogenBond]

Math is only as good as the assumptions it is based on. If the assumptions are not correct, or have not been proven, one can still do the math but the results should be taken with a grain of salt.

Let me give an example, if I was to assume (wrongfully) that gravity is due to the repulsion of matter by space, instead being due to an attraction between matter, one could begin with all the existing gravtiational equations and create a reciprical of sorts to prove my erroneous thesis. The new set of formulas, might be mathematically perfect and could still lead to a excellent correlation that can make predictions, but the math would be out of touch with reality and truth because it is based on an erroneous assumptions.

[questor]

Thanks to Craig and HB for two informative posts. i am not a theoretical or even a '' practicing'' mathematician and i often wonder if math is not the only
''truth'' we posess. i know the definition of an algorithm, but i do not fully understand it. as a matter of fact, i used a system of stock trading for a while based upon algorithms. due to the Random Walk nature of the market, this system was not a big producer of lucre. if math is constantly changing, of what nature is the change? are we developing more sophisticated methods of computation, like a new form of calculus or new types of algorithms? i would think the result would be constant even tho' there may be a new approach to the problem?

[Kriminal99]

Originally Posted by questor

In science, we look for truth. this is frequently difficult to determine and faulty conception can ensue as a result. my question is: can we depend upon math as an immutable truth? are the mathematical predictions of theories such as GTR or quantum mechanics always true, or are there discrepancies?
do we have all the math we will ever need to explain everything, or will perhaps some new ways of computation someday arise? there are many things not yet explained or discovered that have been predicted, what do we need that we don't have? will we find that consciousness, instinct and thought will someday be explained by formulae, or will some things never be
predicted by math?

I dont think so. Math just deals with simpler concepts than logic in general.. such that its less likely that we will make errors in reasoning.

Comparing math to reasoning in general is fairly easy. Every number for example has such similar properties (in fact all numbers can be defined in terms of other numbers) that almost any idea we can apply to one number we can apply to another....

Things other than numbers are a different story... If you think of half a basketball what would that be? A half basketball would be flat so it wouldnt be the same... or would you just define it as having a cover over half so it can still hold air... A "tall hairy goat" is an idea, and a "prime number" is an idea. But a tall hairy goat isn't anywhere as near as well defined which makes it harder to deal with.

[questor]

i don't understand how ''pure math'' can be incorrect. assumptions can be incorrect and computations can be set up improperly, but when the math is done correctly, the results must be reproducible or we could not depend on any computations.

[ughaibu]

The point is that pure maths doesn't necessarilly relate to anything outside itself, so it's correctness is tautological.

[Qfwfq]

Originally Posted by ughaibu

so it's correctness is tautological.

Spot on!

[ronthepon]

Althogh, the very 'branch' of maths called 'approximation methods' is against this. It is also vital at times.

[HydrogenBond]

Math is based on logic, analogous to reasoning. The process is pure, if one plays by the rules of math or logic. The results of good math are also repeatable. The problem that can arise is typcially based on premises and assumptions. If these are off, the reproduceable results of even good math will be consistently erroneous (out of touch with reality). This is where a problem can arise. If everyone is getting the same math result, but is unaware of erroneous assumptions, they may assme that the erronoeus conclusions are correct due to the consistency of the math.

I gave an example earlier of erroneously assuming gravity is due to the repulsion of space. If we were just starting to investigate gravity and lacked the common sense and experience to know this was incorrect, one can still do the math and end up with an excellent correlation. However, although the math will be perfect and reproduceable by others, it could lead to a conclusion that this assumption is reality. The result can be a cascading error, where the math requires more and more assumptions to get it to close. The result can be fantasy instead of reality, yet totally supported by consistent math.