# How useful is mathematics?

1. Oct 23, 2011

### bostonnew

Hi all,

Does someone have a sense of how much of mathematical discoveries eventually are used in applied science?

How long does it typically take for a math paper to be published for it to have an impact on our daily lives?

Do you know of examples of important mathematical discoveries that have never been used for anything outside the mathematical literature?

Thanks

2. Oct 23, 2011

### Number Nine

Mathematics is the language of physics, engineering, chemistry and economics. It also finds uses in subfields of many other disciplines. I don't know if that's useful enough for you.

Of course, the applications are entirely beside the point. Mathematics is interesting enough in its own right.

Mathematics in general isn't concerned with applications. As a specific example, I would point to the Poincare Conjecture.

3. Oct 24, 2011

### Ivan92

Sadly, most people don't care about math let alone the sciences. As Number Nine said, math is the language of the sciences. Without math, we would not be able to prove things in the sciences. Math is more theoretical than anything, so I would not believe that there has been a paper on math that impacted everyday lives.

4. Oct 24, 2011

### jackmell

I think you're missing the forest for the trees but that takes a lifetime to see. It's not just applying math to the sciences. It's something much, much more: our world is full of puzzling things, things we don't understand and I don't mean just science but virtually anything in life, things you might feel math has nothing to do with. Say love. Funny how people behave. Why? Odd how things are in the world, Nature, cultures, life, everything. Many, many people live their lives in a cloud of puzzling haze about the world and that causes a lot of grief in their lives: things happen and they don't understand (really) why. Why doesn't she like me? Why am I not able to do that job? Why did I get cancer? Why? Why? Why? But to know (basically) why, gives comfort and peace of mind.

I believe math explains many puzzling phenomena about our world, much more than just science but I won't go into the details except say I believe all the secrets of the Universe can be found in Differential Equations. And that gives me great peace of mind. :)

And here's some empirical grounding for my assertions:

$$\frac{dy}{dt}=a+by-y^3$$

You'd be surprised to learn how many phenomena in life, not just science, that the principle behind that equation describes. It's under "Catastrophe Theory" and once you recognize things in life descriable by that concept, you'll say to yourself, "yeah, that's why. That's exactly why it happen that way."

Last edited: Oct 24, 2011
5. Oct 24, 2011

### spamiam

I strongly disagree with this claim. http://arxiv.org/abs/0906.0997" [Broken] about using algebra in multiresolution analysis, which has applications in data compression. These aren't breakthroughs on the scale of the Poincaré Conjecture, but they are papers written by professional mathematicians. Moreover, these are just two mathematicians whom I happen to know of--there are undoubtedly many other similar papers.

Furthermore, it's not just a theorem that matters--its proof may contain techniques that others may use in some far-removed application. http://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences" [Broken] is a link about the "unreasonable effectiveness of mathematics" in science.

All these "math is great, who cares about applications?" responses don't seem to answer the OP's question. This view is just an opinion, and not one shared by everybody.

I hope somebody will provide a more complete answer regarding how long it takes for a concept in a paper to be incorporated into an innovation for everyday items.

Last edited by a moderator: May 5, 2017
6. Oct 24, 2011

### homeomorphic

"Math" is too broad a word for this question to be meaningful. If you refer to something like calculus, it's so obviously useful (in a practical engineering sense) that you would have to have your head buried pretty deep in the sand not to realize it.

A more useful question is whether advanced math is useful. That's one that requires a more subtle answer. Especially research-level math. I suspect that a lot of research that is being done may never have any practical significance. But it's hard to predict what will and what won't. Completely non-obvious applications could always be lurking, invisible to the casual observer and even the not so casual observer. Who would have ever thought number theory would be good for anything? Turns out, it plays a role in internet security and other things:

http://www.nature.com/news/2010/100324/full/news.2010.150.html

No one could have had the slightest suspicion that there would be any such applications 100 years ago.

So, it may require a feat of genius to determine whether some math is useful or not. It's not some question that can be given a casual answer.

Sometimes, math provides answers to questions that are nice to know, even if they are not completely essential. If you are doing some modeling and a quintic equation comes up, mathematicians have figured out that you're best bet is probably a numerical method because it's impossible to solve by radicals. I mean, maybe you'd get by without knowing that. You might opt for a numerical solution anyway, but it's kind of nice to know, I think.

Another thing to consider here is that math is very interconnected. No branch of math exists in isolation. One branch of mathematics is connected to many other branches. So, it's all like one big thing, really. You can't really pick out the useful branches because they drag in a lot of other math with them. Sometimes, as soon as one thing becomes useful, it can open the floodgates to many other things that are associated to them. For example, when physicists realized that a gauge theory is really concerned with connections on fiber-bundles, all of the sudden, if that's useful, then potentially, there's a lot of algebraic topology that could come in that is used to study bundles--all kinds of things become potentially relevant.

One practical use for even impractical math is as a way to train the mind. If I gave up math and did something else, you can bet all the math I studied will affect my daily life because it will have changed the way I think. Constantly struggling with problems and theories that are hard enough that you have to get stuck and mull them over will definitely change the way you think.

Finally, math research is cheap, and therefore the bar for usefulness doesn't have to be very high. I don't believe in math for its own sake. Maybe as a hobby, but it wasn't useful, it shouldn't be a profession that people devote their lives to. But I think it is potentially useful, even if the influence is mostly indirect and if it's hit and miss.

7. Oct 25, 2011

### Deveno

one of the things one has to keep in mind, with discussions of this kind, is that abstract mathematics, even something with no forseeable aplication to some physical problem, is still a reflection of people thinking about problems.

any human endeavor that involves as large a number of participants as mathematics does, is relevant, if for no other reason, than as part of our culture. and the fact is, we DON'T KNOW what problems we may face in the future, and we would be foolish to prematurely write-off what a lot of obviously intelligent and imaginative people come up with. the methods developed could very well be far more important than the "results".

mathematics provides us with two visions of the world: insight and out-sight. no one ever argues about the utility of the "out-sight", but you know what? i bet the guy who actually builds a better mousetrap day-dreamt about it, first.

i mean, don't you think that the inventors of calculus knew about zeno's paradox? now, everybody looked at newton and liebniz and thought: "oh how awesome!" (ok, except for that priestly guy), but i'd be willing to bet that more than a few greeks thought: "zeno? what a nutter! who cares about "splitting time", i've got some star-charts to plot."

8. Oct 26, 2011

### Ivan92

My answer isn't to be taken as fact, more it is to be an opinion based on my surroundings and experiences. When I mean "everyday lives", I mean the life of an average person (who does not deal with math or science). I know math can be applied to many different applications to improve tech/science and do agree that it has an impact on the world, but not everyone's life (especially everyday). You do not see everyday people talking about such papers, proofs, or theorems in that matter.

Last edited by a moderator: May 5, 2017
9. Oct 26, 2011

### Deveno

nor do you see "everyday people" talking about engineering issues, or "everyday people" discussing chiaroscuro techniques...unless you happen to be one of those people in YOUR "everyday".

this idea that "everyday" life or the "real world" is somehow divorced from the "theoretical world" that scientists and mathematicians inhabit is just so much poppycock. it's insulting to mathematicians, and it's insulting to "everyday people". yes, any field you care to mention has its own arcane secrets, things that "most" people do not concern themselves with, but that has nothing to do with mathematics, but just with the fact that everyone can't learn everything, all the time.

the hidden agenda behind most discussions of this kind is: "not relevant to me (or straw man substitute), therefore: not relevant". wake up and smell the coffee, limited relevance, or unknown relevance is not the same as NO relevance.

but, even granted some hypothetical "average person" (who by stipulation is not a scientist or engineer, or actuary, or banker, or accountant, or carpenter, or mathematics instructor , or....gee, i hope we have some actual people left, by the time we're done excluding all these folks), the utility of mathematics is so pervasive, we have even devised a special language for it (called arithmetic, or if you're sufficiently educated, algebra) which is deemed necessary knowledge by almost every educational institution on the planet.

and furthermore, even if you are not formally acquainted with any form of the propositional calculus, should you display a basic ignorance of its underlying principles, you will never be taken seriously by anyone who does (yes, if you make an illogical argument, people will laugh at you).

i will make the bold claim that the line between mathematical theory, and mathematical application is so vague, as to be non-existent. the fact that some people are not interested in math, does not mean it's uninteresting.

furthermore, the idea that something ought to have pragmatic value in order to have intrinsic value is such horse manure. we have an entire entertainment industry existing in direct defiance of such a notion (not to mention the world of sports). human beings are NOT tools, existing in order to "get useful stuff done". to reduce the worth of an endeavor to its utility, is to take a shallow and limited view of the possibilities inherent in life. here's hoping those of you who think in such a fashion fail to breed.

10. Oct 26, 2011

### homeomorphic

There are only a tiny number of tribal people left in the world who do not deal with math or science in the sense that they do not use technology that was made possible by it. People drive cars, use computers, etc. A lot of the math used might be pretty old (older than, say, 50 years). But, I just cited a mathematician who is still alive whose work is used in internet security, which people use all the time. Your original post, under any reasonable interpretation, would include these kinds of things.

But let's consider the question of whether most people who are non-scientists or mathematicians use math. Indirectly, they probably do. Assuming their classes were any good, they should have learned a lot of problem-solving skills doing math. 99% of what is useful for most people is just arithmetic--just elementary school-level stuff. But just because people aren't using it doesn't mean they COULDN'T use it. Just because you aren't a scientist doesn't mean you aren't interested in science and understanding the world around you. Actually, that should be part of your duty as a member of society. How can you understand political issues like global warming if you have no understanding of science? It may be that people do not care about such things, but that is a problem. It's not that it's not useful. It's just that people are too lazy to take advantage of its usefulness.

But, there's more to it than utility. Maybe people should actually try to LIKE math. That's a lot to ask, given how badly it is normally taught. But it's really a very interesting subject. The trouble is, 90% of the population just has no idea what math is really about. They don't even have the concept that math can actually give you an aesthetic experience, just like looking at a piece of art.

11. Oct 26, 2011

### Ivan92

Everyone uses math from simple arithmetic to maybe more complicated math. I am not arguing this. What I am arguing is that people will not care about a new paper on math if it will not affect their lives granted that they are not scientists, mathematicians, or technicians. Does everyone use Calculus though? Maybe some Differential Eqs? No, but scientists/mathematicians/engineers do. Thus that is why these kinds of papers will appeal to us rather than the general populus.

No really now? Of course you will be not be taken seriously by someone who is knowledgeable in this field. I know plenty of people who have never taken Calculus and they wouldn't know what the speedometer represents (whether average or instantaneous speed). Yet with out this knowledge their lives go on. Ignorant about the underlying principles of Calculus? Yes. Would they care if they don't think about Calculus? nope. There is nothing we can do. about this.

I am in no way saying that math is useless in our modern world, it is vital and the fact that you cited a paper in how division algebras is used in wireless communication supports this. The math I am referring to is like math beyond simple algebra and arithmetic. What I am saying actually is that no one every really cares about math (lets say beyond Calculus) However the math beyond Calculus is what drives technology and science forward. I agree with what you say actually. People are just too lazy to know these kinds of things or take time to appreciate it.

100% agree to this statement. That's why most people do not care about such math papers (like the ones you posted).

12. Oct 26, 2011

### fleazo

I think the impact mathematics has is often indirect. Go into computer science, physics, chemistry, etc., and see how much influence mathematics have. It is the absolute building block of research published in many fields.

For example, I am writing a thesis on game theory right now. Doesn't seem like math would be directly involved right? But it is, in fact my research would be nothing without it.

The thing about the "obscure mathematics" you are probably picturing as research and wondering what kind of effect it has - well, it expands our current knowledge base of mathematics. The more we know - the more we can use - the more we can prove in any number of subjects.

Seriously, go find an IEEE paper at random, see how much math is involved. You might be shocked!

And beyond its usefulness, does it really matter if you love it? I am not studying math because of its usefulness to be honest. Perhaps that is selfish of me. I study it because it makes me happy. It makes me understand the world better. It makes me understand myself better. It challenges me to not give into the temptation of assumption without proof - it challenges me to look at the world critically and without bias. It is a beautiful and fascinating world and my life is better for it! If you love it, go for it!

13. Oct 27, 2011

### Locrian

It seems like almost once a month I discuss an analysis I produced and someone asks something like "why does an X% change in this result in a Y% change in margin?" The answer almost always involves a derivative and/or a taylor expansion.

You're not supposed to need tools like that for something as "simple" as discussing a multibillion dollar financial statement, but they are a tremendous help and the fact that I can answer those questions when others cannot gets attention. It's easy stuff, but most people can't do it, and often those that can don't think to.

This random, tangentially related thought brought to you by my lunch hour.

14. Nov 2, 2011