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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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I think math makes lots of sense, but only when you relate it to math.. Math is all in relation to itself, a simple example is that the value "x" obviously means nothing in real life except the letter in a language. Even that is abstract so technically nothing makes sense.. Too broad of a question.
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Oh yeah, that was a typo, I meant to have n for all of them but I accidentally put 2 because that was the highest degree for which it did have non-trivial solutions. Do you really think there's mathematical intuition? I don't know how to describe it, but I (and I think many others) feel that there's a difference between say physics and mathematics.
One example would be a field. You could leave it simple as a formula, that given x,y,z you will get some kind of a,b,c. But if you call those a,b,c acceleration and you call x,y,z position and say that all these infinite numbers make up a "field" it feels better. It's just a bunch of words, but these words feel good. This appears to go on until quantum mechanics. I think many have heard some of the greatest physicists say quantum mechanics makes no sense. It works, it's accurate, it's true but somehow those people make a distinction between that and say, thermodynamics.
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Math is everywhere, as Numbers will have you believe.
There are insects with a dormancy cycle that lasts a prime number of years. This has been beneficial to them because while they sleep for 16 years then wake up, breed, and go back to sleep on the 17th, allowing them to avoid parasites. Because the prime number is unfactorable, any parasite would have to have a 17 year dormancy cycle as well, and be in the same phase to prey on these insects.
The whole idea behind math isn't 2 + 2 = 4, it's proving theorems from small building blocks. It is incredibly abstract, but it has had quite a good track record thus far in explaining phenomena.
Edit: Sorry, I've ignored answering whether math makes sense. The answer depends entirely on what you mean by "makes sense." If you mean something that you can understand, math may not make sense, especially in the more esoteric disciplines. It is logically rigourous though, so you won't run into any nonsense in math.
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There are some math things that make sense to me. I know they make sense, because I can't describe it in words, but KNOW that they make sense.
As an analogy, it's kind of like always reading about how to ride a bike and doing all sorts of research on how to ride one but when you KNOW how to ride a bike, no explanation is needed. No words can describe that feeling of KNOWing. For me, that's kind of how certain math concepts feel. Makes total sense.
I think everyone will find some source of "sense" that makes most sense to them to draw analogies from. Often, it's not math. When thinking about doing something new like snowboarding you'll probably draw analogies from skateboarding if you ever skateboarded because skateboarding is what makes sense. If even skateboarding doesn't make sense, you'll have to make do with your experience in walking, because it's the lowest level activity that makes sense without you even having to question why.
Different people with different persuasions will find different sources of sense. I know that when I'm thinking of writing an essay I have to draw analogies from math because math is where I get my source of sense. Yes -- I think math (basically, extremely logically) when I write essays as a result. But I know that there are essay-writing beasts that will just, on one go and without thinking too much, write an essay that would run laps around my essay. For them, writing just makes sense, and I'm sure they don't find themselves asking why. They probably use that as a source of sense and everything that doesn't make sense to them (maybe math) they will try to connect to writing.
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I remember there was some famous mathematician who prided himself on doing math for math's sake, i.e. doing math with no practical applications. I forget the name, but he lived awhile ago, but I would imagine he would be disappointed that many of the things he discovered now have practical applications. It turns out most things in math can be applied to the real world in some way or another, especially in the science fields.
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math was like the only thing that ever seemed to make sense in high school. haha.
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I know that Frege tried to prove that math (or specifically arithmetic) is based purely on logic and their operators (he later tried to extend this to philosophical logic), but his fundamental premise that rested on old class theory failed.
It was later revised by Russell's ramification theory or something, which I haven't read yet, but that was dismissed as being too convoluted and in the end can't be natural logic.
Currently, everything is based on modern set theory, but even that's fundamentals are dubious in that there is the possibility of psychologistic input in it or something...
I dunno, I'm taking a class on this now, and I'm having a hard time keeping up...which is probably mainly due to Frege's mediocre writing style, and the terrible translations that I'm being forced to read as a result.
But in any case, in short, arithmetic and even philosophical logic has no purely logical/rational basis. However, there is currently a neo-Fregean movement that is seeking to re-explore his goals and to try to fix that fundamental flaw in his theories.
I was depressed for two days after learning that...):
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Math makes sense unless it's accounting or bistromatics... then numbers start disappearing everywhere.
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On February 26 2009 15:43 4thHatchery wrote:Show nested quote +On February 26 2009 15:38 alphafuzard wrote:
I remember there was some famous mathematician who prided himself on doing math for math's sake, i.e. doing math with no practical applications. I forget the name, but he lived awhile ago, but I would imagine he would be disappointed that many of the things he discovered now have practical applications. It turns out most things in math can be applied to the real world in some way or another, especially in the science fields. That was Hardy talking about number theory, I think.
Hardy rings a bell
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How could it make no sense if it is one of the science that has the longest history and its application is in almost every single aspect of daily life
It depends on your sense in math that it makes sense or not
e.g (this is from a popular joke), asking 2+2 to different people you may get different answers depending on their "sense" .
mathematician: 4 + ridiculous look
accountant: 4 +/- x percent
economist: everything you want it to equal
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I think you don't understand pure math.
Applied math is a branch of math that has real world application, and is motivated by physics, engineering, etc.
Pure math, on the other hand, explores mathematical ideas simply for the beauty of understanding it. If you approach those problems you listed (prime numbers, polynomials, etc), as you would approach something like art, you might see it differently. Mathematical proofs have a certain aesthetic, and I think those who study math see the elegance in them.
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On February 26 2009 16:35 pavement ist rad wrote:
I think you don't understand pure math.
Applied math is a branch of math that has real world application, and is motivated by physics, engineering, etc.
Pure math, on the other hand, explores mathematical ideas simply for the beauty of understanding it. If you approach those problems you listed (prime numbers, polynomials, etc), as you would approach something like art, you might see it differently. Mathematical proofs have a certain aesthetic, and I think those who study math see the elegance in them.
QFT
Also, there is definetly mathematical intuition.
I'm sure everyone has some on some level.
For a simple example, if you multiply two big numbers togeher your intuition tells you that you will get a bigger one.
You can see why it should be that way, but oly because you have multiplied a lot of numbers. If you later work with other mathematical objects you can develop intuition about them too.
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math makes sence, i need it every day.
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On February 26 2009 16:55 2goons1probe wrote:![[image loading]](http://imgs.xkcd.com/comics/purity.png)
haha nice
I know a friend who is deciding on whether to major in sociology or electrical engineering
i might show her that comic strip =D
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On February 26 2009 16:40 Zortch wrote:Show nested quote +On February 26 2009 16:35 pavement ist rad wrote:
I think you don't understand pure math.
Applied math is a branch of math that has real world application, and is motivated by physics, engineering, etc.
Pure math, on the other hand, explores mathematical ideas simply for the beauty of understanding it. If you approach those problems you listed (prime numbers, polynomials, etc), as you would approach something like art, you might see it differently. Mathematical proofs have a certain aesthetic, and I think those who study math see the elegance in them. QFT Also, there is definetly mathematical intuition. I'm sure everyone has some on some level. For a simple example, if you multiply two big numbers togeher your intuition tells you that you will get a bigger one. You can see why it should be that way, but oly because you have multiplied a lot of numbers. If you later work with other mathematical objects you can develop intuition about them too.
Doesn't that scare you though?
I don't know much about math, but I like to think that math, considering what it is, has a logical basis free of psychologicism...to think otherwise is frightening in its implications, imo. T_T
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On February 26 2009 16:40 Zortch wrote:Show nested quote +On February 26 2009 16:35 pavement ist rad wrote:
I think you don't understand pure math.
Applied math is a branch of math that has real world application, and is motivated by physics, engineering, etc.
Pure math, on the other hand, explores mathematical ideas simply for the beauty of understanding it. If you approach those problems you listed (prime numbers, polynomials, etc), as you would approach something like art, you might see it differently. Mathematical proofs have a certain aesthetic, and I think those who study math see the elegance in them. QFT Also, there is definetly mathematical intuition. I'm sure everyone has some on some level. For a simple example, if you multiply two big numbers togeher your intuition tells you that you will get a bigger one. You can see why it should be that way, but oly because you have multiplied a lot of numbers. If you later work with other mathematical objects you can develop intuition about them too.
I wouldn´t call it intuition, that´s just how our brain works, we compute these things subconciously. Some famous mathematicians commented on their thought process similarly. Consciousness is just a (small) part of our "mind".
Math definitely relates to the real world, our brains are part of nature and every single thought a mathematician ever had was produced/calculated by the neurons in his/her brain, a part of the universe. I am sorry for all the purists out there, but mathematics and the world are interwoven.
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does real life make sense?
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Math makes more sense than any other topic because it the result of pairing logic with (generally) accepted assumptions. Math isn't just a formalism with possible application; it is the language of the universe.
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On February 26 2009 17:57 PH wrote:Show nested quote +On February 26 2009 16:40 Zortch wrote:On February 26 2009 16:35 pavement ist rad wrote:
I think you don't understand pure math.
Applied math is a branch of math that has real world application, and is motivated by physics, engineering, etc.
Pure math, on the other hand, explores mathematical ideas simply for the beauty of understanding it. If you approach those problems you listed (prime numbers, polynomials, etc), as you would approach something like art, you might see it differently. Mathematical proofs have a certain aesthetic, and I think those who study math see the elegance in them. QFT Also, there is definetly mathematical intuition. I'm sure everyone has some on some level. For a simple example, if you multiply two big numbers togeher your intuition tells you that you will get a bigger one. You can see why it should be that way, but oly because you have multiplied a lot of numbers. If you later work with other mathematical objects you can develop intuition about them too. Doesn't that scare you though? I don't know much about math, but I like to think that math, considering what it is, has a logical basis free of psychologicism...to think otherwise is frightening in its implications, imo. T_T
It's not that maths stems purely from people's intuitions. A good intuition will help you explore and understand certain concepts but you still have to prove them using solid theory.
There is a kind of "opinion" in maths. In that it depends on what axioms you're starting with. But almost everyone has agreed to use certain axioms because they work well and correspond with what happens in the observable world.
In that sense it is possible to create an entirely different mathematics that is still logical and works, but no one else is going to agree to follow your definitions. There's nothing to be afraid of.
I don't like when people say mathematics/logic is how our universe works. It's more like the language people use to interpret the universe we live in.
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On February 26 2009 14:38 freelander wrote:Show nested quote +On February 26 2009 14:37 Boblion wrote:
Most of algebra is like astrology :> ??
Cabalistic methods + content completly unrelated to real world Ftw
/rant
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If it didn't make any sense I don't know how the hell are those satellites and space shuttles/stations with people in it up there.
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Calgary25981 Posts
Yes, math makes sense.
No, you don't make sense.
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On February 27 2009 00:14 EsX_Raptor wrote:
If it didn't make any sense I don't know how the hell are those satellites and space shuttles/stations with people in it up there.
Pure mathematicians are too challenged to build satellites and space stations. They evolved in several more adapted species ranging from technicians to engineers and physicians.
o,o
+ Show Spoiler + ok i stop to bash math, i swear :D
But seriously when you talk about real world with a math teacher it is like if you made a blasphemous remark :>
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On February 27 2009 01:04 Boblion wrote:Show nested quote +On February 27 2009 00:14 EsX_Raptor wrote:
If it didn't make any sense I don't know how the hell are those satellites and space shuttles/stations with people in it up there. Pure mathematicians are too challenged to build satellites and space stations. They evolved in several more adapted species ranging from technicians to engineers and physicians. o,o + Show Spoiler + ok i stop to bash math, i swear :D
But seriously when you talk about real world with a math teacher it is like if you made a blasphemous remark :>
I am fairly sure that physicians have nothing to do with space shuttles being fired towards the moon...
Anyway a lot of the maths development throughout history was made by physicists simply because it was needed for new physics discoveries to be made. Like distribution theory and calculus are both based on physics.
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If you can prove math doesn't make sense, good for you, bad for the mathematicians.
If you can't, shut up 
Seriously though, math goes a lot deeper than what most people think. Many mathematical theories have strong physical analogies (check the wiki article on the Riemann hypothesis and the current project to prove it with a physical experiment). I will concede much of modern math is way beyond anything you could ever observe in our reality, and is just plain made up. It's still all based on the same axioms as the rest of maths and as such stands valid and has its own value.
Long story short, "does math make sense" is the wrong question. The right question would be "did you look into it thoroughly enough to make the decision if math makes sense to you".
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On February 26 2009 23:03 Nytefish wrote:Show nested quote +On February 26 2009 17:57 PH wrote:On February 26 2009 16:40 Zortch wrote:On February 26 2009 16:35 pavement ist rad wrote:
I think you don't understand pure math.
Applied math is a branch of math that has real world application, and is motivated by physics, engineering, etc.
Pure math, on the other hand, explores mathematical ideas simply for the beauty of understanding it. If you approach those problems you listed (prime numbers, polynomials, etc), as you would approach something like art, you might see it differently. Mathematical proofs have a certain aesthetic, and I think those who study math see the elegance in them. QFT Also, there is definetly mathematical intuition. I'm sure everyone has some on some level. For a simple example, if you multiply two big numbers togeher your intuition tells you that you will get a bigger one. You can see why it should be that way, but oly because you have multiplied a lot of numbers. If you later work with other mathematical objects you can develop intuition about them too. Doesn't that scare you though? I don't know much about math, but I like to think that math, considering what it is, has a logical basis free of psychologicism...to think otherwise is frightening in its implications, imo. T_T It's not that maths stems purely from people's intuitions. A good intuition will help you explore and understand certain concepts but you still have to prove them using solid theory. There is a kind of "opinion" in maths. In that it depends on what axioms you're starting with. But almost everyone has agreed to use certain axioms because they work well and correspond with what happens in the observable world. In that sense it is possible to create an entirely different mathematics that is still logical and works, but no one else is going to agree to follow your definitions. There's nothing to be afraid of. I don't like when people say mathematics/logic is how our universe works. It's more like the language people use to interpret the universe we live in.
Axioms are essentially irrefutable facts that have were somehow developed and have been empirically proven over time, no? I study philosophy, not math, so I could be mistaken concerning just about everything I say here...but I thought something that has to do with modern set theory or something or other proved most of the old axioms? I at least know for certain that Fregean logic proved Peano's axioms, but I don't really know anything past that...
In any case, math, before you get into the ridiculously theoretical stuff and at least in part, can be seen as the way in which we can comprehend the inner workings of the universe, no? So long as it continually works, it is true. If that is the case, it may be somewhat fallacious but still safe to make the jump that the universe runs on these kinds of mechanics that are mathematically reducible, no?
However, from there, math being irreducible to pure logic (free of psychologicism) is a scary thought in terms of pondering the metaphyics of the universe, is it not?
I dunno, that's just what I've gathered from my limited foray into this subject area.
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math makes sense relative to human beings.
multiplicity is a function of our brains. it is a tool for perception.
multiplicity does not necessarily exist in reality.
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Basically computer came out of mathsmatics, if you want some practical applications. How the internet works, how the cpu works, and all the underlying hardware depend heavily on math. Computer IS made because people wanted a "function generating machine", also a concept out of math.
and so on...
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Math is the most intimate, artificial things we have. If you think about it... If you live in another universe, as long as you're human, math would make sense to you
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Math needs science to give it meaning. Otherwise, it's just a made up extremely boring universe with made up rules. Science anchors math to reality.
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On February 27 2009 13:03 arcticStorm wrote:
Math needs science to give it meaning. Otherwise, it's just a made up extremely boring universe with made up rules. Science anchors math to reality.
i find it's the reverse, just because that's what i've been told.
math anchors science..? no?
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On February 27 2009 13:09 imperfect wrote:Show nested quote +On February 27 2009 13:03 arcticStorm wrote:
Math needs science to give it meaning. Otherwise, it's just a made up extremely boring universe with made up rules. Science anchors math to reality. i find it's the reverse, just because that's what i've been told. math anchors science..? no?
Experiments anchor all. They are what other things depend on. The fact that they can be described by math just means that nature is objective (or something like that).
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On February 27 2009 14:09 fight_or_flight wrote:Show nested quote +On February 27 2009 13:09 imperfect wrote:On February 27 2009 13:03 arcticStorm wrote:
Math needs science to give it meaning. Otherwise, it's just a made up extremely boring universe with made up rules. Science anchors math to reality. i find it's the reverse, just because that's what i've been told. math anchors science..? no? Experiments anchor all. They are what other things depend on. The fact that they can be described by math just means that nature is objective (or something like that).
the funny thing is that wave-particle duality then goes and gives evidence that it actually isn't what we are testing that is objective, it is our relationship to what we are testing.
not to say the relationship of consciousness and matter isn't part of what constitutes "nature".
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On February 26 2009 15:01 Zortch wrote:
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.
The answer.
Math makes sense insofar as the axioms you start with make sense.
With think that Euclidean axioms, for example, are intuitive/common sense. But math can make sense without them.
We just have to sigh and carry on with our calculations. We know that the symbols carry great power and have tremendous pragmatic import as well as theoretical consistency. Underneath that, we don't know. We just can't peel the world open sometimes you know. Like a mango that isn't ripe.
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On February 27 2009 13:09 imperfect wrote:Show nested quote +On February 27 2009 13:03 arcticStorm wrote:
Math needs science to give it meaning. Otherwise, it's just a made up extremely boring universe with made up rules. Science anchors math to reality. i find it's the reverse, just because that's what i've been told. math anchors science..? no?
Think about the example. 1+1 = 2. Wouldn't it be perfectly possible to invent a mathematical system in which 1 + 1 = 0? By why is that we don't say 1+1 = 0? It's because in the real world when we take one thing and put it with another you get 2. Descriptions of the real world (science) is the context from which we understand mathematical concepts. Multiplication is meaningless until we look at it from, for example the distance = rate* time relation. Science is applied Math, but Math is pointless without science.
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Math is an art and alone the quest for beauty and insight is enough reason to do math.
But since it's so useful math is often misunderstood as a mere tool for science (as displayed in this thread), which is a shame.
Math is our method to understand and describe structurs and behaviour between structurs. If our universe is structured in a way we can understand then some mathematical structure is bound to represent or approximate the real world. Even if you dismiss everything else math is about you have to acknowledge math as the most fundamental research you can do.
As always when people who don't know math try to find an example for why math is pointless your example is badly chosen arcticStorm: The structure where 1+1=0 is a very simple one (a so called finite group) and finite groups are basically everywhere around us, like in a clock where 6+6=12=0. Think about it and you will find representations of 1, + and 0 in the real world where 1+1=0 makes perfect sense.
[EDIT]: And as always with fundamental research: you don't know if the results will be useful.
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EPT
