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I know there are many 'math people' on TL and just in general, very intelligent people. So I feel like I can address this question and get some answers. Do you think math makes sense and why?
On some level, I must say that it's inarguable that math makes sense. Things like 2+2 = 4 are just such a basic part of our life that we can't deny it. Everyone knows intuitively his own 'mean value theorem'. And certainly math unfailingly works.
But a lot of math is just math. Prime numbers come to mind. Proving, for example, that there is an infinite number of primes (a fairly trivial proof) is completely unconnected to the real world. Yes, they're used in cryptography, but they're not rooted in any reality except a mathematical reality. Moreover, they go on to to permeate some fields of math and you have theorems about their 'density' or length of their arithmetic progressions.
The most striking example IMO is Fermat's Last Theorem. Simple statement--that there are no non-trivial solutions to a^n+b^n=c^n for n>2--but the proof is a long-ass paper using some insanely crazy mathematics and some guy spent 10 years on it. Even if you can prove it, does it make any sense?
  
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Baa?21243 Posts
It makes sense. You never known when some random math tidbit leads to the next big practical discovery - always happens.
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yes it does
advanced physics sometimes needs such crazy mathematical apparatus
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Most of algebra is like astrology :>
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On February 26 2009 14:37 Boblion wrote:
Most of algebra is like astrology :>
??
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What do you mean by 'makes sense'?
Do you mean that it makes sense from a "real world" perspective?
If so, I don't really think so.
I'm in my third year of math and some of the stuff is super abstract. But as you study math it makes sense in terms of mathematics. So to me, I think math makes sense (more sense than pretty well anything else). Do I think is can always be understood in terms of the real world? Not always.
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It makes sense if you're smart enough and take the time to comprehend stuff. However the way we teach advanced calculus to people like me who are 16-18 years old in high school is a bit silly because I'm pretty certain less than 1% can actually make sense of what is going on. We simply figure out how to do stuff for the test and then forget...
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On February 26 2009 14:33 Hippopotamus wrote:
there are no non-trivial solutions to a^n+b^2=c^2 for n>2
What does "non-trivial" mean? I'm not familiar with that term, but it's not like i'm a math person or anything either.
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Godel's Incompleteness Theorem is relevant to this discussion, but I've never formally learned philosophy, so I can't introduce it in a relevant way.
Oh, a better way to word is is integer solutions, nontrivial is not a rigorous term.
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Well, I guess I enjoy that for the rest of the real world we tend to be able to concisely answer "Why?". Whether or not our answer to that question is correct or unique is a different matter. In mathematics though, the "why" doesn't work the same way.
Recalling something simple like high school algebra, the well-known quadratic formula 'makes sense' because deriving it is analogous to putting a quadratic equation into a "factor squared" form and the solving for x. These are still just words, and this still leaves unanswered why factoring an equation makes sense in the first place, but I suppose it's quite satisfying and it always has been since the day one learned it. But then consider the question "why does integral of 1/x diverge?". There are several ways to show this, but not one of them is satisfying. The proof surely follows from the premises and the manipulations, but somehow there is nothing that "makes sense", there is no analogy.
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Non-trivial depends on the context.
In this case I think its just that none of a,b,c are 0.
Since if you allow them to be zero there are obviously solutions.
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I think what you are asking is fairly simple: maths makes sense because math is the language of the human reason. The basis of all math work (at least everything i know) is logic <> reason <> sense. There are millions of math researchs that dont make sense, but that is because theres not a physics application to that knowledge.
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You can always answer "Why?" in math. Just maybe not in the sense some would like.
You can break any mathematical statement down to te axioms (stuff we assume for doing math). Generally set theory, and maybe a bit more stuff depending.
In the OP you said 2+2=4 is just obvious, but really in formal mathematics from the axoims of set theory you can prove the existence of the empty set and we call that 0, and then the set containing the empty set is 1 and the set containing the set containing the empty set is 2 etc. So now you have the natural numbers, and you can define addition in terms of the successor function (just goes to the next one) and it all 'makes sense' based on our reasonable assumptions.
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I'm not sure what you mean by asking if math makes sense.
I will say that all of the complicated formalisms and rigor is absolutely relevant.
Rocket scientists will casually use physics formulas and complicated integrals to make their designs work. How can they be so sure that they're not making any mistakes? An integral might be "just an integral" to any typical calculus student, but it takes a fair amount of formalism and "density" and real analysis to prove that integrals "will work" 100% of the time. Why is the area under the curve y = x give x^2/2? What exactly is area, and under what conditions can we not integrate? Are there some regions on the x-axis where the line y = x will not integrate to x^2/2? Is the slope always 1? Can I use these results in building my spaceship? This isn't some kind of a managerial science where we can tolerate failures.
Encryption is another example that comes to mind. The RSA algorithm takes a key and hides it in a huge number by multiplying two humongous prime numbers. The RSA algorithm is incredibly secure because there are no known methods to factor huge integers except by brute force. It's the de facto standard in many secure transactions today, and the owners of the patent made billions of dollars creating a company to provide security. The thing that allows the RSA algorithm to even work requires a lot of number theory to prove. Now how can anyone be REALLY sure that it's not easy to factor a large number? It can actually be proven using complexity theory that it IS hard to factor numbers. VERY hard (but not too hard, actually -- once quantum computers come out, RSA will be decryptable ez). Unlike engineering guarantees "this car will run for 10 years! buy this fridge, it will never break!" mathematical guarantees are 100% because of the massive, massive, massive, strong foundations.
And it's this kind of strong, doesn't-make-sense abstract measure topological algebraic geometry foundation that allows us to build cars, planes, RSA security, nuclear weapons, etc.
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On February 26 2009 14:53 Hippopotamus wrote:
Well, I guess I enjoy that for the rest of the real world we tend to be able to concisely answer "Why?". Whether or not our answer to that question is correct or unique is a different matter. In mathematics though, the "why" doesn't work the same way.
Recalling something simple like high school algebra, the well-known quadratic formula 'makes sense' because deriving it is analogous to putting a quadratic equation into a "factor squared" form and the solving for x. These are still just words, and this still leaves unanswered why factoring an equation makes sense in the first place, but I suppose it's quite satisfying and it always has been since the day one learned it. But then consider the question "why does integral of 1/x diverge?". There are several ways to show this, but not one of them is satisfying. The proof surely follows from the premises and the manipulations, but somehow there is nothing that "makes sense", there is no analogy.
In this case, it is a lack of intuition you have for the mathematical concepts at hand. My professor says this pretty often -- sometimes you can read through a proof and understand every single step, even to the end, but the theorem just might not make sense. It's because you haven't developed an intuition for it and you just don't "see" at-a-glance the truth. There are many ways of showing truth, but in the end you really have to see it yourself.
There are certainly analogies -- just none that would make sense to you.
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Canada7170 Posts
On February 26 2009 14:46 Jonoman92 wrote:Show nested quote +On February 26 2009 14:33 Hippopotamus wrote:
there are no non-trivial solutions to a^n+b^2=c^2 for n>2 What does "non-trivial" mean? I'm not familiar with that term, but it's not like i'm a math person or anything either.
a = 0, b = 0, c = 0 is the a trivial solution.
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Actually I'm pretty sure they have no idea if it really is hard to factor large numbers.
Its an assumption (seems to be a good one).
On February 26 2009 14:50 Avidkeystamper wrote:
Godel's Incompleteness Theorem is relevant to this discussion, but I've never formally learned philosophy, so I can't introduce it in a relevant way.
Oh, a better way to word is is integer solutions, nontrivial is not a rigorous term.
Godel's Incompleteness Theorem basically states (very very loosely, I'm not sure on the specifics but this is the gist) that no matter what set of axoims you start with there will always be a statement that you can't prove to be true or false.
We call these statements undecidable.
So they are often thrown in as other axoims if they are things that we think should be true.
For example, the axoim of choice is very common.
It is impossible to prove or disprove that if you have an infinite (countable) collection of sets then you can choose one element from each set.
But it seems so reasonable that math mathematics assumes it to be true as an axoim and it turns out to come of a fair bit.
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Also, Fermat's Last Theorem is
a^n + b^n = c^n
has no nontrivial solutions for n > 2.
The exponents are all n.
Depending on n, there are different families of trivial solutions. If n is odd, the trivial solutions are (0, k, k) for all integers k and all permutations. If n is even, the trivial solutions are (0, k, k), (0, -k, k) for all integers k and all permutations.
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EPT
