The United States Court of Appeals for the Federal Circuit just rejected a patent essentially on the basis that the method used was mathematics. Hooray, they finally almost understood this point, but then they blotted their copybook by seeming to say that it would be OK to patent if the mathematics is hard. Oh dear. Well, baby steps I suppose.

Whenever I hear people put the words “hard” and “math” in the same sentence, an old Einstein quote comes to mind:

“Do not worry about your difficulties in Mathematics. I can assure you mine are still greater.”

Now Einstein was a mathematical genius, but what he said was perfectly true. It is simply that his discipline required such advanced math that most people wouldn’t understand it.

And now they’re suggesting that a judge who may not even be able to balance a chequebook should be deciding what is easy and hard math. In one sense it is like asking a judge what is an obvious or non-obvious invention, except that so many people are prejudiced against math that it would be easy for a lawyer to just that common mathematical techniques are difficult and non-obvious.

Actually, Einstein wasn’t joking he really wasn’t good at mathematics or algebra. Much of his work had to be double checked by his at the time wife who he later left. He just and an inquisitive mind and reasoned through his ideas well but fundamentally wasn’t that fond of the field of mathematics itself.

It is a baseless myth that Maria Mileva checked Einstein’s work. She was a very mediocre student who failed numerous mathematics and physics subjects at the Polytechnic. She even repeated a year. Einstein consistently obtained exceptional grades.

Einstein was being extremely modest. He meant that he worked very hard not that he found mathematics particularly difficult.

Einstein was a mathematics “4-minute miler”. Definitely not the the very best in the world but exceptionally capable anyway.

Einstein’s own contemporaries such as Max Planck and Marie Curie considered him to be a mathematician of the highest order.

And imagine, if he had only patented his stuff…

An interesting point – my liking of GPL is that it attempts to bring an academic type model to software development. A model that has been so successful in science.

I think all this talk of freedom often gets in the way

Actually, Einstein was not good at math. The reason he made great progress, was because all the math he needed was already developed, by Hilbert et al. That math was regarding higher dimensions and invented just by curiosity, they did not see the need for any application of that esoteric math. Einstein just used the math that was already invented, he just applied it. He was no mathematician. If you are good at applying math, you are no mathematician. A good mathematician is able to expand and invent new math that was previously unknown.

To derive new previously unknown relationships in physics – is not mathematical research. It is just applied math, and you are doing physical research. If an economist are using known math to derive new relationships and new economic formulas – then he is not doing mathematical work. He is just applying old math and he is doing economical research.

Today, super string theory has great problems because the researchers need math which is not developed yet. Simultaneously, they need to develop the physics and math at the same time. The math is not done. If all the math already was developed then super string theory would progress much faster.

The super string researcher Ed Witten is the only physics researcher that has done major mathematical progress – he is the only one that has received the Fields medal. As such, he has done lots of progress in physics, and also been expanding the mathematical boundaries. He is a good mathematician, and a good physics researcher. Fields medal normally always goes to mathematicians, and never earlier has anyone else received it (than a mathematician). An economist has certainly not the knowledge to expand on new math frontier, but he is good at applying known math.

Very well written.

But there’s one fundamental flaw: you cannot invent math. That is like inventing a fact. “I invent that the sky is blue.” Math is just an expression of reality, a grasp of the truth, a provable statement.

You cannot patent the grave of Tutankhamun.

You cannot patent evolution.

You cannot patent gravity.

You cannot patent math.

You discover it.

One of my friends finished his maths PhD last year. He presented a paper at a seminar. At the end of the presentation a couple of Russian mathematicians said that virtually identical work had already been done 30 years ago in Russia.

I think that is really cool and shows the strenght of the mathematical paradigm. Sooner or later, you can always recreate mathematical discoveries in an identical way.

There IS a truth in math. True or false. No things in between. No ambiguites as in other sciences.

No, there is prior art ðŸ™‚

Well, some people agree with you, others dont. Some say math is not a science, it is just a language which science uses. I dont agree with that viewpoint, and consider it silly.

It depends on who you ask.

Yes, some people don’t agree with me when I say that math is just a way to describe the truth.

It makes me think of the xkcd strip: http://xkcd.com/263/

A totally ridiculous argument.

By your illogical reasoning George Bernard Shaw was a lousy author because he didn’t invent hundreds of new words.

Practical science invariably develops decades or even centuries ahead of the theoretical explanation.

Michael Faraday’s practical work on electricity science was at least 50 years ahead of the theoreticians.

The modern theory of heat as work was discovered by John Prescott Joule a brewer. He disproved the existing theoretical Caloric concept of heat by practical experiment.

The mechanism by which salicylates (aspirin) relieves pain was only discovered in the 1990s

– thousands of years after willow bark was first used.

Metallurgists only recently rediscovered the technology of making Damascus steel perfected by the Arabs over 1000 years ago.

Physics is applied mathematics. Why would someone invent a new technique if the existing methods were perfectly adequate?

However to say that Einstein was not highly proficient at using existing mathematics is absolute nonsense. Max Talmey, a medical student who boarded with the Einstein family, noted that young Albert was teaching himself analytical geometry proofs at age 11. This is hardly a sign of a mathematical dunce.

Maybe superstring theory isn’t progressing because (like much of modern theoretical physics) it is nonsensical philosopical BS masquerading as science?

Well, I think it is you that is illogical here, because your analogy with Bernard Shaw is illogical.

I am debating whether Einstein was a good mathematician: did he invent new mathematics? No. Did he invent new physics? Yes. So, Einstein did not invent new math. This is true. He was an excellent practitioner of math.

Regarding Bernard Shaw. I am not discussing whether Shaw was a good language researcher, if he… dont know, but maybe solved some tricky grammatical research problem. Found new rules or so that explained some ambiguities in linguistics. No, instead Shaw was a good AUTHOR, not a Language Researcher. He was like Einstein: good at using the tools that was already there: the english language. Shaw did not expand on linguistics, but prose.

I hope you agree that Shaw was a good author, but (possibly) a bad researcher in linguistics.

Einstein: used the tools very excellent. The tool was math. But Einstein did not expand on the tool itself: math.

Shaw: the tool was the english language. But Shaw did not expand on the tool itself: english.

If you are good mathematician, you can give new mathematical knowledge. Or you are just a physics researcher.

I have not claimed that Einstein was bad at USING math. He was good at it. But, he did not do any math research. He did not expand on the boundaries of mathematics. He expanded on the boundaries of physics.

You have it backwards. String theory has great problems because we’ve been unable to experiment and observe mathematical predictions.

We’ve reached a point where we have mathematical theories built on top of mathematical theories and few of them have backed up with repeatable experiments. Thus these theories work on the basis of assumptions (ie assuming the theory of the xyz is correct then taking that research further).

This is why bigger and every more expensive equipment (such as the LHC) are being built: to try and observe many of the predictions that mathematical theory has described.

So yes, Super String Theory has it’s problems, but not because of a lack of mathematical theory, it’s because we’ve been unable to experiment against it to weed out the incorrect theories from the potentially correct ones.

Cool. I heard Ed Witten and some other super string theorists say they have problems because the math is not invented yet, on a TV program.

If the physicists can verify the experiment or not, that is another question. Some people (many mathematicians) does not really think that it is too important to verify the theory. Many mathematicians thinks that the invention of new math that is totally useless – is the biggest achievement that can be done.

Some math chair man (Hilbert?) said in a toast “may pure math NEVER find any application in the real world!” and every mathematician gladly toasted.

For many mathematicians, the big thing is to play intellectually, with abstract entities. It is not important whether it is of any use or can be verified. The intellectual satisfaction is enough.

Unfortunately, much math finds applications in the real world, later. Number theory was considered as the most pure form of math, because it had no applications. But later, number theory is used in cryptography. It has been polluted. Gauss, the greatest mathematician ever, said “Math is the Queen of sciences, and number theory is the King of math” – or something similar.

A lot of the maths isn’t yet. But that’s because most of the maths hasn’t been tested either.

There’s several multitudes of mathematical theories relating to quantum mechanics, string theory, holographic theory and so forth. None of them complete and many of them are incompatible.

Thus it is impossible to finalise any maths in their field without first identifying the equations that are wrong – and this cannot be done without testing via experimentation.

I wasn’t aware of this. In fact I was always lead to believe the opposite was true; that mathematicians were constantly trying to create mathematical proofs: http://en.wikipedia.org/wiki/Mathematical_proof

This often leads to the invention of new mathematics but it’s far cry from the untested scenario that you describe (in fact I’m pretty sure that mathematic theorems undergo the same level of peer-review as the sciences do – so any errors in the proofs are usually identified)

I’m sorry but I find that view point completely backwards.

It’s a bit like saying “lets invent the best motor vehicle known to man” then telling everyone they have to walk everywhere out of principle for keeping motor vehicles an intellectually pure concept.

I’m all for furthering our understanding from an intellectual stand point, but that’s because I know full well that such knowledge trickles down into mainstream applications and, for the most part, benefits everyones lives on a daily basis (whether it’s easier communication to distant loved ones, more secure banking or even just more immersive video games)

I’d like to think while most mathematicians might joke about keeping their creations pure, they are happy – even proud – to see their creations improving peoples lives. However I’d challenge any mathematician that doesn’t feel that way to give up all of his luxuries and try living life from the perspective of someone who cannot benefit from applied mathematics. It would be interesting to see how long they last

Edited 2011-08-22 16:35 UTCThere is no need to finalize math via experimentation. Math is about theory. Not experimentation. Lot of math has never had any connection to reality. But still mathematicians pursued that direction. And finally, 50 years later that math became practical.

Many mathematicians frown upon experimentation. If you know you are right – then you are right. No need to test it.

I am not contradicting your paragraph here. I am just saying that mathematicians strive to create theory and expand math borders. That is always done via proofs. Without a proof – you have achieved nothing. So yes, mathematicians strive to create proofs. That is how math is done.

Crudely, you can say:

The finest of sciences is math. The finest fields in math – is pure math. That is, math that has no application in real life.

[/q]

Pure math that has no application – is nobler than applied math. This is what many mathematicians say. You will never see any mathematician abandon a discovery he made, because it did not have any application. Instead, he will maybe be able to publish something in Pure math – which is really something to strive for.

If you want to apply math, become an engineer. Or physicist. Then pure math is not for you.

And no, I am not joking. Talk to math researchers, or read books – for instance “Fermats last theorem” by Simon Singh – and you will see I am not joking on this.

Edited 2011-08-22 17:12 UTCWe’re talking about string theory though, not pure mathematics. Thus that changes everything. Science is tested though observation and experimentation. That’s just how it works.

You’re missing my point. I’m not arguing against maths for maths sake (in fact I actually praised that in an earlier post).

I’m just say that it’s wrong to say that mathematicians don’t care if their maths is correct and gave reasons to support that claim.

Again you’re missing my point. I’m well aware of what pure maths is and I’m well versed in Fermat (and not just his lost theorem / Andrew Wiles proof, which is pretty much pop-culture these days) . I’m not saying mathematicians shouldn’t be focused on pure maths with no application. In fact I’m all for that. I’m just saying that I highly doubt that most mathematicians would never want to see a practical application of their work – or rather that they would some how begrudge another individual using that principle for applied work.

It’s one thing to state that mathematicians don’t care about applied work (that’s something i think we all agree on), however it’d a whole other leap to say that they begrudge having their work tainted with applied derivatives – which is what an earlier post (in my opinion, wrongly) stated.

So while I’m sure they might joke about it (just as I’m sure we all make snooty comments about things we look down upon), I’m sure they no more hate seeing their work put to practical use than we literally hate things we joke about.

Maybe i’m wrong. I don’t know any pure mathematicians. I just find it hard to believe that anyone that bright would be that short sighted.

Edited 2011-08-22 21:13 UTCI dont agree. Math is also science, and there is no need of experimentation in math.

Of course mathematicians are obsessed with correctness, that is why math is done via proofs – to get it correct. But, they dont care if the math is verifiable in real life. That does not mean they dont care about correctness – they do.

Oh for crying out f–king loud – there /IS/ a distinction between science and pure mathematics. You raised the point about string theory and I said that a lot of the theories need testing. Yes pure mathematics doesn’t need experimentation (peer-review and proofing are the equivalent), however the sciences do.

It’s all very good and well saying that string theory is applied mathematics, but if you’re applying said maths to a literal, physical (albeit in a 10/11/26 -depending on the version of string theory you conform to- dimensional space) property or object, then it’s no good just saying “the maths works”, you then need to make predictions based upon that theory which then need to be observed. Like it or not, that

how the theoretical sciences work. After all, if a theory is true, then itisWILLbe observable.The whole point of physics is about learning the world around us. It just so happens that the world is best described in the language of mathematics. However that doesn’t mean that physics -like the other sciences- doesn’t work on the principle of predictions, observations and repeatable experimentation.

If you’re still not convinced this is the case, well then I suggest you pick up a few physics books and find out for yourself

Glad we agree on at least on thing

Edited 2011-08-22 22:08 UTCOh for crying out f–king loud – there /IS/ a distinction between science and pure mathematics. You raised the point about string theory and I said that a lot of the theories need testing. Yes pure mathematics doesn’t need experimentation (peer-review and proofing are the equivalent), however the sciences do.

It’s all very good and well saying that string theory is applied mathematics, but if you’re applying said maths to a literal, physical (albeit in a 10/11/26 -depending on the version of string theory you conform to- dimensional space) property or object, then it’s no good just saying “the maths works”, you then need to make predictions based upon that theory which then need to be observed. Like it or not, that

how the theoretical sciences work. After all, if a theory is true, then itisWILLbe observable.The whole point of physics is about learning the world around us. It just so happens that the world is best described in the language of mathematics. However that doesn’t mean that physics -like the other sciences- doesn’t work on the principle of predictions, observations and repeatable experimentation.

If you’re still not convinced this is the case, well then I suggest you pick up a few physics books and find out for yourself [/q]

I have studied some pure physics courses at uni, but not specialized in it. I also, for instance, studied Fourier solution of heat equation, etc. I was interested in how to solve such an PDE, not in the physics. Of course the heat equation has a physical interpretation, but we did not really focused on that. We were just studying the Fourier method on whatever problem crossed our road. And we examined the boundaries of the Fourier method. etc.

But regarding if Math is a science or not. Some say yes, others say no. I think math is science. The difference to me, is that math is a deductive science, whereas all other sciences are inductive.

Inductive science: I see only white swans, hence all bird swans are white. I have established a fact from many observations: all swans on earth are white.

I see that stones always fall to the ground, hence, there is something that draws stones to the ground on every planet in the universe.

Deductive science: No observation is made. We dont relate to the physical world. Instead, we invent a set of rules and can manipulate the rules formally. And then we see what the rules lead us to. Coincidentally, lot of these rules happens to have a counterpart in real life – that is why the rules where chosen.

Often we choose silly rules that are apparently wrong (let us assume a triangle sums to less than 180 degrees, or, a triangle sums to more than 180). That is silly. However, it turned out to be very fruitful and did have practical applications (relativity theory). Should we no pursue this research because the rules were silly? No, then Einstein would have had problems later.

Thus, the difference is that when we have established a fact in math: it is true. Forever. It will never happen that someone says “hey, I discovered that pythagorean theorem is not valid in a euclidean space”. No, we have PROVED correctness. Hence it is true. Pythagoreas were true 2000 years ago, and will be true in another 2000 years. I will never be false. It has been PROVED in an abstract setting. In a well defined universe, with some rules – pythagoreas will always be true.

However, there exist black swans on earth. Just because you do lots of observations – does not make anything true. Einstein is correct today. But, in 2000 years, will Einstein be false? 2000 years ago, the physics where not true. But math 2000 years ago is true, even today and will always be true. Math never changes. When established, it is true forever. THIS is TRUE science – to me.

In physics, biology, economics, etc – things might change. Things that were true 2000 years ago, will not be true today. But math is. Math is true forever – THAT is science. The rest of the sciences are… I dont know. But not true facts. Subjective, not objective.

Hence, I consider Math a science that deals with true things. Forever true. Whereas physics deal with things that might not be true in 2000 years. That is not really a science – to me.

I think you’re still missing the point of physics.

In physics, you still need a mathematical formula for it to be considered complete. However mathematical formulas have to correlate with observations as well.

Lets go back to Fermats last Theorem: The equivalent in pure mathematics would be the proof being the maths and testing a sample of

an + bn = cn(please excuse the lack of superscript) formula as observation. The fact that we know a sample of those formulas always come out true would support the proof and visa versa.In physics it’s all very good and well applying pure mathematics to a problem, but we don’t know all the facts and thus are only working on a limited understanding. This means that often the maths is wrong because it doesn’t factor in something critical. Hence why observations are required to collaborate the mathematical predictions.

An example of this point would be Newtonian physics: from an 18C perspective his formula was bulletproof. However Einstein came along and proved that such calculations didn’t factor in space curvature nor time. Then with quantum mechanics came a whole new set of rules that is stranger than anyone could have predicted. And now we have a number of competing string theories, holographic planes and a whole host of other fringe theories and interpretations.

In many of these cases the maths works, but it’s based on assumptions that we can’t validate so each physicist therefor believes their own maths is correct and everyone else is wrong. Clearly everyone can’t be right, thus we need observations and experiments to disprove the wrong ones.

The math is not wrong. It is the physics that is wrong. If something has been mathematically proved, then it is true.

If it turns out that physicists did not incorporate something, then the math is still correct – but it is the physics that is wrong.

However, Kurt GÃƒÂ¶del applied math, on math itself to find the boundaries of math. And he found astonishing things – there is limit in every axiom system. He proved this 1931.

Some time later, the physicist Bell came along and proved Bell’s Theorem. I consider it one of the greatest achievements in the 20th century. He basically proved that “the world as we see it, is wrong”. Our picture of the world, is wrong. Something is terribly wrong and we dont really know what.

He said: either the statistical forecastings by quantum mechanics is wrong OR the locality principle is wrong. Both can not be true. It has been proven that the statistical forecastings by quantum mechanics is true. Hence, the locality principle is wrong.

If something happens in the Andromeda Galaxy, it WILL affect us. Someway. We dont know how. (Maybe quantum entanglement can explain that?). Earlier we that things happening in another galaxy, far away will not affect us – the locality principle. Which is wrong.

Kebabbert,

“The math is not wrong. It is the physics that is wrong. If something has been mathematically proved, then it is true. If it turns out that physicists did not incorporate something, then the math is still correct – but it is the physics that is wrong. ”

You know that this comes off as an incredibly arrogant perspective? Math can be mathematically correct and still be wrong in terms of being an accurate abstraction of what it’s meant to represent. Maybe this is actually what you are saying, but it was unclear.

“He said: either the statistical forecastings by quantum mechanics is wrong OR the locality principle is wrong. Both can not be true”

Quantum entanglement is interesting. Logically either the two particles share a similar state when they are entangled to produce identical events, or they instantly communicate the events at a distance.

“It has been proven that the statistical forecastings by quantum mechanics is true. Hence, the locality principle is wrong.”

To be pedantic, the statistical forecasting of probabilistic events doesn’t *prove* the absence of an underlying mechanism – we may never find it, but there will always be some legitimate doubt.

I’m aware that very sophisticated experiments have failed to show any correlation. However we know that any finite sequence of events can be represented by a sufficiently complex algorithm. It’s legitimate to ask if we simply didn’t observe enough events. For example: those very same experiments could have been conducted against a whirlpool hash function instead of quantum events and the researchers would have been unable to find a pattern despite the fact that they were indeed generated deterministically. Even with just 100 bits, the researchers are simply unable crunch these exponential datasets. At best, we can put lower boundaries on the complexity of mechanisms hypothetically needed to produce the events.

Last I heard quantum computing experiments have lost cohesion beyond around 10 qubits. Until our experiments show that quantum mechanics scales well into at least hundreds of qubits, I think there is still room to question whether there is a non-probabilistic mechanism at work.

After that, there will always be the remote possibility that the mechanism is more complex, but it’s more likely that god’s just playing dice with the universe.

“If something happens in the Andromeda Galaxy, it WILL affect us. Someway. We dont know how. (Maybe quantum entanglement can explain that?). Earlier we that things happening in another galaxy, far away will not affect us – the locality principle. Which is wrong.”

I’m just a QM amateur, but this is at odds with my understanding: The quantum events can’t communicate anything about the other galaxy back to us. All we know is that if someone else is reading quantum events on the entangled atoms, then they’ll get the same sequence. As soon as anyone tries to force an atom’s state (say to communicate a message) then they loose entanglement. It’s a good thing too since otherwise they’d violate causality in general relativity while sending messages faster than the speed of light.

Just something else to consider with regards to the quantum non-local event versus internal state debate:

I sort of implied that the events would necessarily be deterministic in terms of a mathematical algorithm. However it’s possible that the internal state be deterministic and non-discrete.

An example of this is the 3 body (or n-body) problem. The ability to predict the outcome is directly proportional to the accuracy of the initial measurements and the precision of subsequent iterations. Eventually any mathematical simulation of the n-body problem with less than infinite precision will diverge from reality. Anything less than infinite precision implies that the simulation will eventually repeat itself after all the finite states have been simulated.

So, in theory, quantum particles could have internal state which is exactly copied during entanglement to produce a deterministic stream, and yet have no mathematical algorithm to accurately predict events, and no way to fully observe the initial state.

I know this flies against quantum theory, however in defense the idea that quantum particles with discrete states cannot have non-quantum mechanics internally to supply the “randomness” is only an assumption.

One principal difference is the exponential super-positioning of quantum bits. An internal state mechanism will not have this property, and thus at some point we would reach it’s mechanical limits. This would convince me of the truth that QM is probabilistic. However to date, physics isn’t advanced enough to make a convincing distinction.

Interestingly enough, even if quantum particles turn out to have a deterministic internal state, they still have valid applications in terms of cryptography. An infinite supply of non-random deterministic events is still ok since there’s still no way to simulate the internal state to find the initial conditions of the entangled atoms.

Edited 2011-08-25 18:04 UTCIt’s often the maths that lead to incorrect theory! Do you even know how theoretical physics works?

I’ve stated a thousand times already that a mathematical formula can balance correctly but it doesn’t mean that it applies to the situation correctly.

Regardless of how your narrow-sighted opinion of maths might argue about the infallibility of numbers, if a formula is written to solve a real world problem and doesn’t, then the maths is wrong. Period.

It’s like using 2+2=4 to work out the acceleration of a falling object. Using your argument the maths is correct, however if it clearly doesn’t solve the question thus it’s completely wrong for this application.

This is why physics needs to be tested.

Also I should add that quite often the mathematics /IS/ wrong. The level of complexity involved is quite literally astronomical. It’s not exactly hard to make mistakes

In fact (and going back to your Fermat reference earlier) Andrew Wiles’ first proof was proved wrong when it had been opened for peer-review. Had to later revise it to the proof we know now.

So your argument isn’t even correct from a pure mathematics stand-point.

Edited 2011-08-25 11:25 UTC> he just applied it.

“just applied” is a strange way to describe it considering that all of his fellow physics researchers were not able to “just apply the maths” as you say.

> If you are good at applying math, you are no mathematician. A good mathematician is able to expand and invent new math that was previously unknown.

So in your definition of mathematician, there is no difference between a mathematician and a mathematician researcher..

> The super string researcher Ed Witten is the only physics researcher that has done major mathematical progress [cut] As such, he has done lots of progress in physics,

No! It will be a progress in physics only when it will be shown that the string theory is a good description of reality, currently it’s only a mathematical progress.

Dont you agree that if you are expanding the boundaries of physics, then you are doing research in physics? An economist that expands the boundaris in economy, is doing research in economics? You can possibly not claim they are doing expanding the boundaries of math!

Exactly. A mathematician do research and expand the boundaries. That is what he does.

The practical use of the formulas that mathematicians invent, is left to the engineers, physicists, economists, etc.

Not necessarily mathematical progress. If the string theory is not expanding on the boundaries of math, then it is not mathematical progress.

The judge has obviously never met a great mental mathematician.

Willem Klein calculated the seventy-third root of a 507-digit number in 2 minutes 43 seconds in his head without pencil or paper. He was the computer at CERN for many years.

http://stepanov.lk.net/mnemo/mkleine.html

As a chip engineer I just assumed “hard” meant as in hardware vs software rather than hard vs easy. So in that respect I agree such a math patent is valid either in hardware or software.

I have a patent that could be implemented in hardware (it was) but could have been just as well done in firmware or software or brass nobs and broomsticks for all I care>

Now I do object to most of the idiotic software patents that are based on usually obvious ideas like one click shopping or the current Apple vs Samsung “looks a lot like mine”. But going against patents because they could be done in software instead of hardware is just stupid. Hardware, firmware and software are just tools to build things.

As an engineer myself, I assumed that the judge meant that “hard” maths was simply maths that was so far beyond a person’s ability to do it with pencil and paper that, in order for the “invention” to be practical at all then the math had to be performed by a computer.

This need not involve higher maths functions at all, it could be as simple as the four basic arithmetic operators, but simply requiring millions of operations per second. A video codec might be an example of what the judge had in mind. This is math that in theory people could perform, but practically nowhere near quickly enough to be able to actually decode compressed video in real time.

“Too hard to be practical, therefore it requires a computer, therefore it is a patentable invention”… this might be the thinking.Some discussion on this decision from a legal perspective can be found here:

http://www.groklaw.net/article.php?story=20110818143312919

At least 99.9% of PhD qualified mathematicians work supporting engineering. physical sciences and economics. At my local university (one of the largest in Australia) the very small maths department basically exists to help teach several thousand engineering students.