New Math?

[Stravo]

I was watching an episode of "Criminal Minds" and at one point when trying to talk down a wacked out physicist the FBI profiler is fawning praise over him and mentions "Some of your string theory analysis can't even be completed because they haven't invented the math yet that can carry out the analysis." My immediate reaction was "Math not invented yet?" I am NOT math inclined but isn't math sort of a way of doing things with numbers, like a language using numbers instead of words. So is it possible that there is a way or formulas that does things with numbers that has not been invented in the last 3,000 years? Seemed a little odd to me.

[Darth Wong]

I was watching an episode of "Criminal Minds" and at one point when trying to talk down a wacked out physicist the FBI profiler is fawning praise over him and mentions "Some of your string theory analysis can't even be completed because they haven't invented the math yet that can carry out the analysis." My immediate reaction was "Math not invented yet?" I am NOT math inclined but isn't math sort of a way of doing things with numbers, like a language using numbers instead of words. So is it possible that there is a way or formulas that does things with numbers that has not been invented in the last 3,000 years? Seemed a little odd to me.

Modern computer encryption techniques were invented in the last few decades, after thousands of years of math. Keep in mind that real, earnest exploration of theoretical math really didn't take off until the last couple of centuries. For most of human history, people thought that addition was pretty advanced math.

[Eleas]

I was 9watching an episode of "Criminal Minds" and at one point when trying to talk down a wacked out physicist the FBI profiler is fawning praise over him and mentions "Some of your string theory analysis can't even be completed because they haven't invented the math yet that can carry out the analysis." My immediate reaction was "Math not invented yet?" I am NOT math inclined but isn't math sort of a way of doing things with numbers, like a language using numbers instead of words. So is it possible that there is a way or formulas that does things with numbers that has not been invented in the last 3,000 years? Seemed a little odd to me.

Why would the field of mathematics not expand? Witness Fermat's last theorem for an example of this. IIRC in the attempts at solving it, mathematicians employed vector mathematics, which in Fermat's age did not exist.

NOVA: Eventually, after a year of work, and after inviting the Cambridge mathematician Richard Taylor to work with you on the error, you managed to repair the proof. The question that everybody asks is this; is your proof the same as Fermat's?

AW: There's no chance of that. Fermat couldn't possibly have had this proof. It's 150 pages long. It's a 20th-century proof. It couldn't have been done in the 19th century, let alone the 17th century. The techniques used in this proof just weren't around in Fermat's time.

Emphasis mine. Source: http://www.pbs.org/wgbh/nova/proof/wiles.html, which I found on the strength of Google-fu. I have no way to verify the article, except that I have heard the same said in other articles in the past. It's fascinating stuff, although of course way, way above my head.

[Eleas]

Modern computer encryption techniques were invented in the last few decades, after thousands of years of math. Keep in mind that real, earnest exploration of theoretical math really didn't take off until the last couple of centuries. For most of human history, people thought that addition was pretty advanced math.

Also, mightn't the existence of computers and applications running on them have spurred the field? I'm talking relatively prosaic fields like digital airflow simulations, virtual nuclear testing, or even 3d computer games. It seems to me like there had been scant need for such mathematical applications before the advent of the computer.

[Gaidin]

A few of my math professors was notorious for insisting that the first two years of the degree were just the introduction and getting our feet wet. To them, the real math was in the theory behind the number applications we typically played with. Which makes sense, to me. I got the impression that even the encryption techniques and other modern applications fell under field and/or group theory where the theory was heavily simplified down into the equations and matrices most people associate with math. Then theres still other fields like topology and set theory.

So yea, as far as math goes, its quite possible to derive and/or invent a new equation within a given field. And if you change the axioms and operations enough, you can totally invent a new theoretical field altogether and start from what is essentially scratch.

[Surlethe]

My immediate reaction was "Math not invented yet?" I am NOT math inclined but isn't math sort of a way of doing things with numbers, like a language using numbers instead of words. So is it possible that there is a way or formulas that does things with numbers that has not been invented in the last 3,000 years? Seemed a little odd to me.

Your understanding of math is slightly simple, which explains your surprise. Math isn't simply a language using numbers instead of words, although that's part of it. Instead, it's the structure you can build by showing some things imply other things. The heart of math is proofs: "If X, then Y".

In some sense, since logic itself doesn't change, math can't be "invented" the way cars or cell phones are. If the existence of the natural numbers implies Fermat's Last Theorem, then it implied Fermat's Last Theorem several minutes after the Big Bang and it will imply Fermat's Last Theorem after the heat death of the universe. But in another sense, we haven't discovered the proper theorems to apply to more esoteric problems -- like some of the ones in string theory, e.g. -- so those theorems have to be "invented" before the problems can be solved.

If you're interested, there are actually many standing problems out there. Here is a list of the more famous ones. Some of them (for example, Goldbach's Conjecture and the Twin Prime Conjecture) have been around for several centuries.

Finally, on a more practical note, if math has been static for millennia, why are faculty members paid to do research into it?

[Ariphaos]

At the beginning of Analysis class, our professor went briefly over the history of the current subject, and mentioned Newton's invention of Calculus, while the current topic was some of Riemann's work. "So we're getting into 19th century math now. Congratulations!"

She was a little weird.

Regardless, there's no shortage of problems that need solving. The development of computers also allows for exploring problems, algorithms, and solutions that simply weren't feasible fifty years ago.

[Eris]

Back to the theme of the OP, we can not only have new maths, but we often use maths before we even know how to fully describe it. Calculus has been used since the time of Newton and Leibniz with spectacular results in physics, chemistry, engineering, and so, but it wasn't until many years after it was first developed it that calculus was characterised with mathematical rigour by Cauchy with the formal definition of a limit. Until then they just fudged and made do.

Mathethamics is a field of research and study like any other. It's an evolutionary process, and one that really only got kick-started relatively recently. A lot of the really ground-breaking work that opened up the field was done in the 18th and 19th centuries.

[Sriad]

My immediate reaction was "Math not invented yet?" I am NOT math inclined but isn't math sort of a way of doing things with numbers, like a language using numbers instead of words. So is it possible that there is a way or formulas that does things with numbers that has not been invented in the last 3,000 years? Seemed a little odd to me.

Your understanding of math is slightly simple, which explains your surprise. Math isn't simply a language using numbers instead of words, although that's part of it. Instead, it's the structure you can build by showing some things imply other things. The heart of math is proofs: "If X, then Y".

In some sense, since logic itself doesn't change, math can't be "invented" the way cars or cell phones are. If the existence of the natural numbers implies Fermat's Last Theorem, then it implied Fermat's Last Theorem several minutes after the Big Bang and it will imply Fermat's Last Theorem after the heat death of the universe. But in another sense, we haven't discovered the proper theorems to apply to more esoteric problems -- like some of the ones in string theory, e.g. -- so those theorems have to be "invented" before the problems can be solved.

If you're interested, there are actually many standing problems out there. Here is a list of the more famous ones. Some of them (for example, Goldbach's Conjecture and the Twin Prime Conjecture) have been around for several centuries.

Finally, on a more practical note, if math has been static for millennia, why are faculty members paid to do research into it?

I always assumed it was like my college of music's Study of Ancient Music department.

(I KID!)

[Durandal]

I was watching an episode of "Criminal Minds" and at one point when trying to talk down a wacked out physicist the FBI profiler is fawning praise over him and mentions "Some of your string theory analysis can't even be completed because they haven't invented the math yet that can carry out the analysis." My immediate reaction was "Math not invented yet?" I am NOT math inclined but isn't math sort of a way of doing things with numbers, like a language using numbers instead of words. So is it possible that there is a way or formulas that does things with numbers that has not been invented in the last 3,000 years? Seemed a little odd to me.

Oh, you'd be surprised at the things you can do with numbers. For example, there are maths that show not all infinities are equal, something that is commonly implied to basic math students. Infinity is infinity is infinity. Not so. There are countable infinite quantities and uncountable ones. And this is a very powerful concept in theoretical computer science.

Since math is the tool by which physicists describe the universe, physics does have to wait for math to catch up on occasion. Quantum mechanics would've been impossible to put together without imaginary numbers. Sure, they're imaginary and don't mean anything, but they're necessary tools to solve the differential equations involved.

Remember that math really isn't about numbers; it's a way of determining truth and falsehood (or degrees thereof, in the case of fuzzy logic-type math). New maths are simply new ways of thinking about problems. Instead of saying that the square root of -1 is undefined, let's just say that it is and we just don't have a numerical representation for it. So we'll call it i and use it to solve some previously unsolvable stuff.

And wouldn't you know it! When we do that, we can solve the equation, and the solution has no negative square roots. Hot damn, them mathematicians are smart.

[Master of Ossus]

More math was been discovered in the 20th century than in all previous centuries combined, at least according to my old math professor. The fact that even most college-level math was discovered long ago leads to the misconception that math doesn't change and advance like other sciences, but in fact it's been developing quite rapidly.

[Surlethe]

More math was been discovered in the 20th century than in all previous centuries combined, at least according to my old math professor. The fact that even most college-level math was discovered long ago leads to the misconception that math doesn't change and advance like other sciences, but in fact it's been developing quite rapidly.

When you study math, you essentially to retrace the steps of all the great mathematicians of the past. You start with simple addition, subtraction, and geometry -- products of the ancient Greeks. Then you move to algebra, which was given to us by the Muslim world during the dark ages. By the time you get to calculus, you're up to Isaac Newton and the 1600s; multivariable calculus and discrete mathematics are 1600s-1700s, analysis and foundational calculus was discovered during the 1800s, and so on. By the time you get to graduate courses, you're moving slowly up through the 1900s.

There's a reason it takes years and years to get your PhD and start actual research: you need to catch up to your field before you can begin advancing it.

[Surlethe]

Regarding the idea that "math is a language of numbers", perhaps, Stravo, you're thinking of how math is used in the sciences. In that case, it's closer to the truth. Scientists (and engineers too, I suppose ...) use math to quantify and describe nature because math is rigorously defined and useful in communicating quantifications objectively. You can think of math as a language through which truths about nature are communicated, but there's a lot more to the structure of math than the stuff that gets applied.

[Stravo]

This has been really enlightening, right off the bat Mike's example of encryption technology was forehead slapping doh moment of obvious new math but the idea of math without numbers which I believe was highlighted in the movie "A Beautiful Mind" just seems so weird to someone like me who took Calculus as his highest level of mathematics during my education. Thanks for the lesson on the subject guys.

[Xenophobe3691]

Oh, you'd be surprised at the things you can do with numbers. For example, there are maths that show not all infinities are equal, something that is commonly implied to basic math students. Infinity is infinity is infinity. Not so. There are countable infinite quantities and uncountable ones. And this is a very powerful concept in theoretical computer science.

As well as in Quantum Physics. Renormalization relies entirely on differing degrees of infinity. Just look at infinite sequences and series.

Since math is the tool by which physicists describe the universe, physics does have to wait for math to catch up on occasion. Quantum mechanics would've been impossible to put together without imaginary numbers. Sure, they're imaginary and don't mean anything, but they're necessary tools to solve the differential equations involved.

Actually, in electronics and vibrations, imaginary numbers really do have meaning. Just like negative numbers can represent debt, and multiplication can represent "groups" of "groups", imaginary numbers are used to represent periodic quantities. The magnitude of the vector is the amplitude, but divided into real and imaginary components, the real component shows the value affecting reality, and the imaginary component determines the sinusoidal behavior of the real component (essentially, the imaginary component negates what would otherwise be the normal value from fully expressing.)