Did Anyone Invent Some New Math Stuff So They Could Solve A Problem Which Then Lead To a Technological Breakthrough?

April 6, 2011 1:16 PM

Can you cite any examples where a technological breakthrough wasn't possible until there was some sort of mathematical breakthrough?

I've been told that Egyptians were able to forge ahead in geometry, which better allowed them to figure out the areas of their fields. I'm not exactly sure how useful this is, but this is the best example I can think of.

I'm also somewhat familiar with Leibniz and Newton and calculus, but did either of them start with a problem to solve, "figure out" integral calculus and then use it to solve their problem?

Did anyone else make a breakthrough once they figured out new mathematical techniques?

posted by Brian Puccio to science & nature (32 answers total)

Asymmetrical encryption is the basis for much of modern secure telecommunications.
posted by unixrat at 1:18 PM on April 6, 2011

Bayesian statistics is the basis for all bayesian filtering systems (e.g. spam filters)
posted by the mad poster! at 1:19 PM on April 6, 2011

Bayesian statistics seems to go back to Thomas Bayes (1702–1761), so it doesn't look like some dude looking to write a spam filter figured out this branch of maths and was able to write a spam filter, it looks like the math was here before the technology. Thanks though!
posted by Brian Puccio at 1:21 PM on April 6, 2011

Boolean Algebra and the computer?
posted by Lutoslawski at 1:28 PM on April 6, 2011

So your actual question is "did anyone, looking to solve a specific problem, come up with new mathematical techniques which allowed them to solve their problem?".
posted by zamboni at 1:29 PM on April 6, 2011 [2 favorites]

Nuclear weapons.
posted by kirkaracha at 1:30 PM on April 6, 2011

Didn't Isaac Newton run into some unsolvable problems with his physics theories, so invented calculus to move forward?
posted by Eyebrows McGee at 1:39 PM on April 6, 2011

I gather that CT scanning required certain math principles in order to accurately construct images emerging from the multiple scans.

from wikipedia:
Tomography has been one of the pillars of radiologic diagnostics until the late 1970s, when the availability of minicomputers and of the transverse axial scanning method – this last due to the work of Godfrey Hounsfield and South African-born Allan McLeod Cormack – gradually supplanted it as the modality of CT. Mathematically, the method is based upon the use of the Radon Transform invented by Johann Radon in 1917. But as Cormack remembered later,[27] he had to find the solution himself since it was only in 1972 that he learned of the work of Radon, by chance.
posted by jasper411 at 1:43 PM on April 6, 2011

Bomb and artillery siting.
posted by StickyCarpet at 1:55 PM on April 6, 2011

E = mc^2
posted by jchaw at 1:58 PM on April 6, 2011

Shannon information theory was a huge innovation that made the internet possible, as well as a pile of other technologies.

From wikipedia:

"Applications of fundamental topics of information theory include lossless data compression (e.g. ZIP files), lossy data compression (e.g. MP3s), and channel coding (e.g. for DSL lines). The field is at the intersection of mathematics, statistics, computer science, physics, neurobiology, and electrical engineering. Its impact has been crucial to the success of the Voyager missions to deep space, the invention of the compact disc, the feasibility of mobile phones, the development of the Internet, the study of linguistics and of human perception, the understanding of black holes, and numerous other fields[citation needed]. Important sub-fields of information theory are source coding, channel coding, algorithmic complexity theory, algorithmic information theory, information-theoretic security, and measures of information."
posted by zug at 2:02 PM on April 6, 2011 [3 favorites]

John von Neumann. Especially his work in computer science and economics--although he was a pioneer in a bunch of fields.
posted by anaelith at 2:09 PM on April 6, 2011

The development of chaos theory has opened a lot of doors.
posted by Chocolate Pickle at 2:11 PM on April 6, 2011

Didn't Isaac Newton run into some unsolvable problems with his physics theories, so invented calculus to move forward?

Sort of. Here's the question, as I understand it:

I roll a ball across a flat surface, and measure the distance it travels after time t_1, t_2, t_3, etc. I want to know the velocity at which it's traveling, but all I can come up with is the average velocity over some length of time (d2-d1)/(t2-t1). Newton wanted to know the instantaneous velocity. He pretty much had to invent calculus to get that. Calculus answers the question "what happens when (t2-t1) gets really, really, really small.

But really, he wasn't starting from a physical problem per se. Rather, he was starting from a foundation of geometry and arithmetic curves.
posted by muddgirl at 2:17 PM on April 6, 2011 [1 favorite]

So your actual question is "did anyone, looking to solve a specific problem, come up with new mathematical techniques which allowed them to solve their problem?".

Exactly. Problem, new maths, breakthrough.

Thanks everyone for your feedback, these are all excellent jumping off points!
posted by Brian Puccio at 2:25 PM on April 6, 2011

did anyone, looking to solve a specific problem, come up with new mathematical techniques which allowed them to solve their problem?

That would be Claude Shannon's Information Theory. He was working at Bell Labs and needed to answer the basic question, "How much information can we reliably pump through one of our wires?"
posted by Chocolate Pickle at 2:29 PM on April 6, 2011 [1 favorite]

It's not exactly what you asked, but Chua's prediction of the memristor is in the ballpark:

Chua’s theory [...] is framed in terms of the basic equations of electric circuits. Those equations link four quantities: voltage (v), current (i), charge (q) and magnetic flux (φ). Each equation establishes a relation between two of these variables. For example, the best-known equation is Ohm’s Law, v=Ri, which says that voltage is proportional to current, with the constant of proportionality given by the resistance R. If a current of i amperes is flowing through a resistance of R ohms, then the voltage measured across the resistance will be v volts. A graph of current versus voltage for an ideal resistor is a straight line whose slope is R.

Equations of the same form but with different pairs of variables describe two more basic electrical properties, capacitance and inductance. And two more equations define current and voltage in terms of charge and flux. That makes a total of five equations, which bring together various pairings of the four variables v, i, q and φ. Chua observed that four things taken two at a time yield six possible combinations, and so a sixth equation could be formulated. The missing equation would connect charge q and magnetic flux φ and would describe a new circuit element, joining the resistor, the capacitor and the inductor. Those three devices had all been known since the 1830s, so the new element would be a very late and unexpected addition to the family. Chua named it the memristor.

No law of physics demanded that such a device exist, but no law forbade it either; the existing theory of circuits with resistance, capacitance and inductance could be augmented in a straightforward way to include memristance as well. Chua argued for the plausibility of the memristor on grounds of symmetry and completeness, suggesting an analogy with Dmitri Mendeleev’s construction of the periodic table. Nature is not required to fill every square of this table, but a blank spot is certainly a good place to look for a new chemical element—or a new circuit element.

posted by harmfulray at 2:38 PM on April 6, 2011 [3 favorites]

Oops, here's the cite for that quote.
posted by harmfulray at 2:41 PM on April 6, 2011

von Neumann studied the Prisoner's Dilemma and things like it in Game Theory in part because he was developing the doctrine of Mutually Assured Destruction as a way of preventing an intercontinental nuclear war.
posted by Chocolate Pickle at 2:56 PM on April 6, 2011 [1 favorite]

I mentioned Chaos Theory above. That's actually an example of what you seek. The guy who came up with the foundation of Chaos theory (the idea of extreme sensitivity to initial conditions) was trying to explain why his computer programs couldn't predict weather a long time into the future. Chaos theory made clear that it isn't possible to predict chaotic systems accurately a long way out because you can never know the initial condition sufficiently accurately. Any error, however infinitesimal, in how you set up your model would cause it to diverge from reality eventually.

It didn't solve his problem, though. What it did was demonstrate that it was impossible to do what he was attempting to do.
posted by Chocolate Pickle at 3:03 PM on April 6, 2011 [1 favorite]

Seconding nuclear weapons.
posted by Blazecock Pileon at 3:10 PM on April 6, 2011

I just thought of another one. John Snow developed what we now would think of as "epidemiology", an application of statistics to analysis of causes and spread of disease, in response to a cholera epidemic in London in the 1850's.

He had available to him a huge listing of cholera deaths, with information about all the victims, and tried slicing the data in all sorts of different ways in order to see if he could find correlations with other things. One thing he did was plot deaths on a map of the city, and in doing so he found one cluster, all centered around a single water pump. He was thus able to show that cholera was being spread by drinking water contaminated with human sewage. (This is all the more noteworthy because his work predated the Germ Theory of disease.)
posted by Chocolate Pickle at 3:14 PM on April 6, 2011 [1 favorite]

Nuclear weapons are definitely not a good example of this at all. Einstein came up with the principle of mass-energy equivalence in 1905, almost 40 years before the Manhattan Project began. At the time, nobody even knew there was such a thing as an atomic nucleus. Furthermore, nuclear fission does not convert matter into energy; it breaks apart atoms to release some of the binding energy that holds the nucleons together. The fact that this energy has a small but measurable mass is beside the point. Nuclear weapons were based on physical research on the properties of atoms, not on any new mathematical breakthroughs.

With a few notable exceptions, like Newton's development of calculus, I think this is more common in fiction than in real life. For instance, in Ursula Le Guin's novel The Dispossessed:

"What is the ansible?"

"It's what he's calling an instantaneous communication device. He says if the temporalists—that's you, of course—will just work out the time-inertia equations, the engineers—that's him—will be able to build the damned thing, test it, and thus incidentally prove the validity of the theory, within months or weeks."

TV Tropes has a page on "formulaic magic" which is a similar idea.
posted by teraflop at 4:00 PM on April 6, 2011

Gosset (better known as Student) developed the T-test for brewing and agricultural work.

Fisher basically developed much of modern statistics for use in population genetics.
posted by benzenedream at 4:27 PM on April 6, 2011

Cracking of the Enigma machine using the Banburismus procedure, which is related to the sequential analysis technique in stats.
posted by scalespace at 10:19 PM on April 6, 2011 [1 favorite]

GPS and relativity?
posted by palindromic at 6:49 AM on April 7, 2011

Again, the theory of relativity was formulated long before GPS was conceived -- probably before such a thing was even imaginable. The question asks for mathematical breakthroughs that happened in response to a technological problem.
posted by teraflop at 7:11 AM on April 7, 2011

I don't know if I'd say that logarithms were invented to make hand-calculations easier, but they were certainly more exploited than they would have been because of that. Like, hand-calculations are hard, logarithms exist and that's good, but then someone is like "but I need to do THIS thing which is SO CLOSE to what we can already do...I think I'll figure out something extra about logs and BLAMMO! now I can do it!"
posted by DU at 11:53 AM on April 7, 2011

Usually the fruits of one field are food for another. A result in one field may be a solution in another. Autotrophic disciplines are few and far between.
posted by TwelveTwo at 3:31 PM on April 7, 2011

That is, it is rare for an answer resulting from a given line of inquiry to also be the very question that is posed to produce the answer in the first place. Instead, you must shift focus and aim at a different problem, then reframe the original problem according to the results borne from the alternative side problem. If the problem doesn't require such a shift then it isn't much of a problem, you can just continue unspooling the consequences out from what is already known. So, yeah, a mathematical result allowing for a technological problem to be solved and a technological result allowing for a mathematical problem to be solved are both just as likely. The absence of such cross-pollination would be the real shocker. It would imply that the discipline is perfectly self-contained and devoid of theoretical problems and gaps.
posted by TwelveTwo at 3:46 PM on April 7, 2011

And to more directly answer the question, a good example is Feynman's path integral. A mathematical tool invented to resolve a physics problem that to this day is still a big of a bugaboo for mathematicians seeking to form a rigorous definition of it. This is a quite a bit different than the survey of existing mathematical techniques and discovering that one or a few of them are immediately applicable to a problem exterior to its origin. That is, the path integral was a mathematical tool that was originally foreign to mathematics, and I so should fit the example you ordered.
posted by TwelveTwo at 3:56 PM on April 7, 2011 [1 favorite]

I'm aware that it's quite rare for the math to not already exist/be known when a problem arises, which is why I came here. Thanks to those of you who came up with exactly what I was looking for.
posted by Brian Puccio at 6:25 AM on April 9, 2011