Why Math?

Just learned that Israel Gelfand is no longer with us. R.I.P.

Try not to take this the wrong way, but I think I love you. Seriously though, I wish there were more like you. I run into people all the time that think learning is hard, even though they can't help but do it. I mean, I'm not good at math, per se, because I don't practice at it in the raw state but at least I understand it's crucial, that the things I do, my memorized functions as it were, are expressible, if not entirely derived, from math.

Terrific post! I'm definitely one of the "but math is hard" crowd - in high school I adored it and was geeky enough to belong to the math club, but undergrad destroyed any pleasure I had in it. I need to find a good book at the popular science level at my local library and see if I can rekindle that enjoyment. Thanks!

My mom is one of those math-phobes who could be good at math. I'm convinced. Now, I haven't caught her taking the determinant of any matrices; I doubt she even does any "everyday" algebra. But her favorite class in school was Logic (in the Philosophy department) and she likes solving sudoku puzzles. Sounds like a mathematical mind to me.

Not that it matters, but I think you are, in a sense, deeply mistaken. In math, you can prove things, that's right. But all your examples come from physics, and that's a different story.
In physics, a working device IS the proof, and a mathematical computation showing that it cannot work, is simply wrong, or perhaps the theory behind the math is.
The Banach-Tarski paradox always comes to my mind. Math says it works but...
Modern physical theories are well tested, but testing is not proving.
In math, you say "if this axiom holds, then" and then you prove.

Math is deductive, but you still need to use induction and observation to demonstrate that a particular mathematical model accurately describes the world.

The impetus behind General Relativity, a beautiful bit of math, came from observation. And contrary observations could disprove it. The math would still be right, though.

When people like me argue that math is ultimately tautological, that's not a smear. Mathematics is useful precisely because it is tautological and thus amenable to proof, and not merely corroboration.

"obfuscatory mathematics"

Redundant!

I was one of those people. Math seemed like pointless work in high school, and no one could explain to me why I should pay attention. Now I've found some strange interest and fascination with it, and I'm working my way through a basic algebra book on my own, anxious to keep learning this 'pointless work.' Huh.

Wonderful post, though. :)

Oddly, mathematics is not tautological, but why appears to be a question for evolutionary biology (all the lineages where brains dominated but the math was wrong got eaten).

Someone above pointed out that a working device is a proof in physics. There's a crucial flaw here. What's it a proof of? That device. Physical principles are relations among symbols in our heads, symbols that we have learned to associate with patterns in the world around us, or other patterns of symbols in our head. The device doesn't care, but any kind of induction requires analysis (that is, cutting up perceived reality in terms of symbols which our brain can manipulate). That is purely mathematical.

Now, the question of making the jump from an analysis in one case to an analysis in general hasn't really gotten beyond where Kant left it, unless we accept a sort of anthropic principle that things which can't adapt to patterns in the world get eaten and don't reproduce. But in that case we head straight for phenomenology, and the intending agent simply has a tendency to intend certain sequences of patterns.

I agree that mathematics is beautiful, and an incredibly powerful tool in trying to understand and predict what will happen in the "physical world". In fact I'm drawn to ideas like those of Max Tegmark's that the "physical world" might in fact be nothing more than pure mathematics in the final analysis.

However, I still think there are some problems with what you've said. For one thing, we do not currently have a 100% accurate and complete theory of how stuff works out there in "reality". At one stage Newton's Laws seemed to be the final word, but now we "know" that they're only an approximation that can be very wrong in certain other more extreme circumstances, and they just don't cover area areas at all. Quantum Mechanics is also potentially wrong or at least incomplete in some way, and so on. I'd love to hear how mathematics can be used to prove or disprove my claim that I'm "conscious" right now.

This kind of objection may seem rather extreme, but I think it pays to remind ourselves that right now we only have imperfect and incomplete mathematical models of how stuff appears to work out there in the "real world". They're the best we have, but they almost certainly aren't perfect. Given that, there is always at least a chance that a purely mathematical argument is actually talking about something slightly different from reality.

Mathematics can't "prove" anything about reality in an absolute sense unless we're also 100% sure that the mathematical model we're using is in fact "the truth, the whole truth, and nothing but the truth" in every last respect. How can we ever be that sure?

On the other hand, if one person says something like "a machine heavier than air can't leave the ground and fly through the air to some distant point and then return to land at the starting pointing again, all under it's own power", and then someone else proceeds to build a working aircraft that you observe with your own senses doing exactly that, then it's pretty hard to say they haven't "proved" it can be done!

There is no mathematics available to us now that can truly model every aspect of the (claimed) faster than the wind cart and ground, etc. Even if we all agree that using Newton's Laws and so on will be "close enough" for our purposes, the actual specific details of any one cart may turn out to be too complicated to be predicted reasonably accurately in anything less than a computer simulation of some kind. It may be possible in some cases to "prove" that a cart (for example) can never exceed the wind speed (or some other limit) using a correct argument based on a high level analysis of total energy or momentum in the system and so on. But it's also clear that sometimes the complexity or the cumulative effect of various small errors introduced by simplifying assumptions in the purely mathematical approach will mean the only way to know if a particular device will actually work properly in the real world is to build it.

I really think that some of the fun comes from the fact that 'math' by itself is an abstract (pure) system that is entirely and totally unrelated to the real world. It is just a set of axioms and formulas. When and where it comes in useful is by relating it back to the real world in some practical way such as physics, economics or other applied maths. In any case we map real elements directly onto abstract ones, perform some types of calculation and then map the answers back to real meanings. A mathematical model of an engine is just exactly that, an abstract model that relates back to the real world.

Pure maths, like group theory, ring theory or graph theory are unencumbered by the messiness of the world around us. They exist in a clean abstract state that allows them to be internally consistent. Oddly, computing models like Turing machines are also in this pure category, although what they compute most often comes from and is effected by the real world.

Mathematics is interesting because although you cannot argue the results, you can always argue with the mappings or interpretation. While the math may be correct, the way it has been applied is always flawed by the real world to some degree (although the degree itself might just be trivial).

What has always made mathematics so special to me is that it can, unlike everything else bound in the real world, achieve a real measure of 'perfection'. Simple arithmetic for example never fails, has no errors, and will always produce the correct answer, no matter what is going on in the world around it. Nothing is cooler :-)

Paul.

milan_va: The Banach-Tarski paradox always comes to my mind. Math says it works but...

That's an interesting example. But you left off something critical, right at the ellipses. Were you about to say it works in math but we know from physics that it doesn't? Because if you do, I will ask you how does physics know that it doesn't work. Would you going to start giving me some mathematical argument? Or are you going to say, well, I tried it with a soccer ball and couldn't get it to work.

I'm not sure I agree with Mark here, b/c reality is reality, and math is not. But I'm not sure how to work my way out of this one...

-kevin

I generally agree. However I think you have a blind spot.

Why do so many people have "terrible math teachers"? Why do we generally not hear "Oh, I loved math, even though not very good. I love to play around with it!" It's fine to try to convince random people that math is good, but how are you going to fix the social / cultural processes that make the experience of math for most so bad?

My own experience is that many math Ph.Ds tend to think that math involves skills that should not be sullied by practicality or accessible explanations. I once ruined a Ph.D candidate's week by explaining that the beautiful lambda calculus that he thought was purely useless had been implemented as a programming language.

I don't have any concrete suggestions for fixing this, but I guess a good (though tiny) first step would be to agree that the mathematicians and the educational system are at least as much deserving of corrective admonitions as the mathphobes and crackpots.

Mark,

While I agree with the gist of what you say, your examples are coming from physics, not mathematics.

By working out the energy inputs and the energy outputs, and comparing them. That's math.

That's not math. Working out the energy inputs and outputs is physics, not math at all. Same thing with:

Math. You can compute the amount of energy required to split the water and compress the gas in an ideal system...

Using your arguments, I could prove that I can go faster than light -- just use Newtonian mechanics. The math would be right, the physics wrong.

Marc

Very nice post.

I too have noticed that many people have a poor understanding of what math *is*. To most, math was the subject that involved a lot of x's in high school. They see it as a system that they don't fully understand, like many other subjects they had in school. What they don't see is that math isn't an invention, its a discovery. It isn't the creation of some set of cute rules, its the expression of natural logic and reasoning in a quantitative way.

I've challenged people who think of math as a limited sub-section of science, "Show me a subject that doesn't involve math." Responses vary, but you get replies like "music" or "people". Because they can't convert something to a number and solve for X, they're tempted to think that it doesn't involve math. But music involves a lot of math -- just sound involves a lot of math, never mind organized sound. People involves math, never mind the functioning of the human brain, look at patterns of human actions. Traffic flows, purchasing decisions, the propagation of fashion trends, etc. (Never mind the concept of a "soul" or "emotions", regardless of what those are it is clear that the resulting actions generate many patterns.)

Most people only see the final plug-n-chug results of math, they never use the analytical or abstraction parts.

Some people come into the world looking for elegance and cohesion. Some just seem to treat it all as a bunch of unrelated facts. The latter tend to misunderstand math.

And I absolutely agree that math is essential to understanding the world around you. Understand logic, understand quantitation (not really a word, but it fits), and you can't meet a problem you can't analyze. You're only limitation is your own brain.

Full agreement on the first part of the article. There seems to be this myth that people believe: you either "get" math, or you don't. End of story. Problem is, in my experience, many math teachers appear to believe this too. They are perfectly capable of explaining math to kids who "get" it, but not to the kids who don't. After all, the teachers were usually the ones who "got" it when they were in school, and can't really imagine what it's like to not instantly get it. I've actually heard math teachers say stuff like "Why don't you just see it?"

The second part I'm a bit more critical about. Sure, if you have a mathematical model of a system, you can analyze it to your heart's content and find out if the system does what you hope it does - if your model is correct. You need to understand the physics before you can even formulate the model, and no math will tell you if you forgot to include an aspect of the system. Math may help you answer the question, but it doesn't help you to determine what question you should ask (even if the answer is not 42).

For the Brown's gas example, for instance, physics tells you to look at the thermodynamic properties of H2, O2 and H2O to figure out the heat released in burning H2 and O2. And it wasn't the math that told you to add another equation, it was yet more thermodynamics that told you to include the effect of the gas expansion. Only after a careful analysis of the physics involved does the math really come in.

Of course, sometimes the math does warn you that your model is incorrect, like when your math gives you unphysical results. But even then it's really the physics that disqualifies your answer.

The problems with perpetual motion machines that I've seen mostly seem to stem from the fact that their inventors don't really understand the physics, not so much that they get the math wrong. In particular they often don't appear to know what they're measuring (in one example I remember they didn't understand how to properly measure power on AC current). They might do all the calculations correctly, but they just won't do the correct calculations.

Inspiring, truly inspiring!

(But I have to agree with some of the comments above, most of the examples you give are in the realm of physics-land)

Many thanks for this post! This is inspiring to mathematicians worldwide, and should be translated in all the languages of the world and engraved in stone in the tympanon of teaching places. Using very small type, of course. Or this should inspire us to build greater teaching places.

Well, at least, this does much, much to raise my morale as a mathematician currently stuck to teaching in high school (and to particularly math-phobic youngsters at that). Thanks.

Re multiple people:

Yes, the examples are all physics - but the point is that you can't do physics without math. Math is deeply entwined into almost everything in one way or another.

For example, one commenter talked about how if relativity were wrong, we'd learn that through observation.

But for relativity to be disprovable, it's necessary that it make specific predictions - which it does with math. And if observations didn't match relativity's prediction, we would know that by using math.

Finally, as I tried to say in the post, math isn't just the abstract stuff. When we create a model with something, the modelling process is still a part of math. Relativity might have beautiful mathematics - but a large part of it is just building a mathematical model of an aspect of reality. If the mathematical model proposed by relativity doesn't match observations, then it's not just physically incorrect, it's also mathematically incorrect.

@4
You comment reminds me of one of my most favorite web comics:

http://xkcd.com/435/

As a chemist in training (who is not as good at upper level maths as he is at lower level maths -_-), I have to agree with you absolutely. When I run my caluclations and optimizations, there is nothing better for proving my point than having a heapload of numbers, some equation connecting the points in a meaningful manner, and those beautiful, beautiful graphs that describe my data to a T. Math is gorgeous. It can be elegant or brutal, simple or complex, but above all else, it is useful and necessary to everyone and everything.

tl;dr. I love math, especially the stuff I just don't get yet.

@11
Well, you can put any clever thing in that three dots ;-) But IMHO only "I've tried and it doesn't work" is the right answer.

@18
Well, physics would be useless without ability to generalize. By definition, using logic ("if this worked, that will work too") is math, so you cannot do physics without math (just used logic myself :-)).

I just wanted to ask you to think of a better formulation next time you say "math proves it can/cannot work". Current physics proves; math shows; math proves if; something like that.

I mostly agree. Before I nit-pick, let me disagree instead with a commenter:

"The impetus behind General Relativity, a beautiful bit of math, came from observation." [#5, boyo]

As I recall, the impetus behind General Relativity (GR) was a mathematical analysis of Special Relativity (SR), not any new physical observations.

Within a year of Saint Albert's 1905 publication of SR, the paradox was pointed out that a rotating bicycle wheel would have the circumference (rim) shrink but the radii (spokes)stay the same length. In that year, half a dozen suggestions were made to patch the paradox. Finally a paper was published which reviewed these, and concluded that, under SR, wheels could not rotate. Einstein agreed.

Einstein began working on GR because of Mathematical problems with SR, or, to put it another way, mathematical analyses of the models available as fitted to old data.

We know that wheels do rotate. So the ratio of circumference to radius of rotating physical disks is not Pi. Why? The warping of space that characterizes GR was explored, and Einstein had to have people tutor him in the appropriate Math (Tensor calculus, which happened to have been invented by Mathematicians for their own abstract reasons).

So the commenter does not, in my humble opinion, properly resolve the metaphysically puzzling relationship between Math and Physics, and his critique of Mark CC thus evaporates.

I agree that there is so much need for blogs that show people that they don't have to be afraid of math!

I am a recovering math-phobe and am blogging about my experience. You might find it interesting to check out since it's related to what you're doing: http://www.nomoremathphobia.com

I've worked as a volunteer math tutor in the public schools (middle and high school) for the past four years. In my opinion, one of the biggest problems with the way most math classes are taught is that they are presenting math as a series of isolated facts and techniques that need to be memorized, but leaving out the parts about how the techniques fit together, why you might want to use them, and how mathematical proof gives us a firm basis for understanding parts of the world we live in. A lot of this is teaching to the test. Kids learn about manipulating fractional exponents because it's going to be on the standardized tests, and then later they learn a few rules about manipulating logarithms, but they never have explained how these two things are connected.

You can compute the amount of energy required to split the water and compress the gas in an ideal system, where the compressors and valves and electrodes and such are all perfect and lossless. And still, the amount of energy required in that perfect lossless system to split water and compress the resulting gases will be greater that the amount of energy extracted by burning them.

In the ideal system, wouldn't it be equal?

Re: MarkCC @ 18

Yes, the examples are all physics - but the point is that you can't do physics without math.

Perhaps this is just semantics, but isn't a toddler "doing physics without math" when he takes his first steps? How about a flying mosquito?

In any case, if what we call "reality" is in fact absolutely nothing more than a manifestation of pure mathematics, then you might also say that mathematics is the only thing that truly exists and so (in a sense) is the only thing we can ever truly "do" no matter how hard we might try to do something else!

Re the XKCD comic mentioned earlier, I like to think that the big gap between "physicists" and "mathematicians" should be filled with something like "universe constructors"!

But for relativity to be disprovable, it's necessary that it make specific predictions - which it does with math. And if observations didn't match relativity's prediction, we would know that by using math.

I would also say that relativity could also be disproved simply by the observation of any faster than light phenomenon that could be used to transmit information. That would effectively show that one of the axioms used in "relativity" (a particular mathematical model of some aspects of "reality") was in fact not true in that same "reality".

Math is a language. One that doesn't translate well into English, or other "natural" languages, but one that is perfectly suited for describing how the universe works (at many levels).

If you want to talk about the natural world, you need to resort to the only language that can describe it - Math.

So, yes, MarkCCs examples are from Physics, but the point is that Math is the only language that you can describe Physics accurately in. English just isn't good enough (or Mandarin for that matter).

Mark,

I agree with the sentiment and overarcing message of your post, but I think you've unintentionally trampled on the philosophical underpinnings of science. Statements of the form "___ is impossible." strike me as deeply unscientific and dogmatic. We may have accumulated staggering numbers of observations which suggest that creating a perpetual motion machine is overwhelmingly unlikely, but strictly speaking, I can't think of any way to scientifically demonstrate the impossibility of any and all such devices. Certainly, we can falsify specific perpetual motion theories with counter examples, and we can identify contradictions with current theories. But, the moment we start treating conservation laws (eg. energy, momentum, constant speed of light) and other scientific principles as infallible god-given truths, it seems to me that we've left science proper behind and entered into some sort of religious discussion. I always understood science to be fundamentally skeptical. In fact, I think one of the great scientific strengths of your blog is how it encourages critical thought and a skeptical perspective.

I also think the other posters made some particularly good points regarding the distinction between physics (more generally the empirical sciences) and mathematics. I'm not aware of any mathematician who requires empirical--especially physical--verification or justification to establish the truth of a mathematical proposition. Similarly, I'm not aware of any physicist who doesn't require empirical justification or verification to establish the truth of a scientific proposition. This is not to say that math isn't incredibly useful and necessary to phrase scientific propositions, just that it is always insufficient, on its own, to establish their truth.

Finally, your comment, "If the mathematical model proposed by relativity doesn't match observations, then it's not just physically incorrect, it's also mathematically incorrect." reminded me of knot theory and its confused history. Formal study of the subject was initiated by Peter Guthrie Tait, based on Lord Kelvin's idea that matter was fundamentally composed of knotted vortices swirling in the aether. As a result, Tait compiled tables of all knots using 10 or fewer crossings. These tables were intended to serve the role of our modern periodic table of elements. Of course, the aether theory was debunked and the knotted vortices model was thus proven "physically incorrect". However, the mathematics involved remained entirely sound (modulo a few logical errors). Although the physical failure of the knotted vortices model certainly detracted from the utility of nascent knot theory, it did nothing to damage the correctness of the mathematical theory.

First, I am a mathematician.

The Banach-Tarski paradox is only non-physical because the proof requires a continuum--and since real matter is made of atoms, and not divisible to infinite fineness, it doesn't apply to real matter. The assumption is stated in the hypothesis--and that is part of why it is a THEOREM--not an antimony or even a paradox.

Math is NOT a language--and I am getting tired of the meme that math is a language ( clearly invented by some liberal artist). Math CONTAINS
a language.

Math, often at a very high level, underlies ALL of technology, and all of science. Remove the math, and what is left is very primitive.
Without the Radon Transform--no MRI CAT; without the Patterson Projection--no structure of DNA; without Calculus, no physics.

On the other hand, mathematics is its own thing, mathematicians do not exist to serve applied science or other sciences, and we know what we are doing--as we are the most successful "scientific field"
in terms of research progress currently. Even more so than Biology--just look at the famous math problems of the 20th century
that have been solved--things like the Poincare Conjecture.

Many people are math phobes for several reasons:

FIrst and formost--math is hard, and doesn't allow bullshit.
If you are interested in easy verbal baloney, false arguments,
appeal to authority etc, DO NOT TRY TO BECOME A MATHEMATICIAN--you will be properly cut to shreds.

Second, math is largely taught at the elementary level by non-mathematicians without a clue what math really is. It is NOT
a language, it is not a bunch of procedures for solving essentially rote application problems. Math is a creative subject, with the
intention of ABSOLUTE logical rigour. It has to make complete sense,
and is evolutionary from it's foundations. Proofs need to be taught from the GETGO--yes, even in elementary arithmetic. Otherwise, you are not learning MATH: You are learning "monkey tricks".

Third: Most people will never understand math. We must accept that.
It requires a very high level of intelligence and a special talent, and there is a lot of it--which is why mathematicians specialize in one or two or three small fields of math--and work like dogs.

Yet, as I have said, it underlies all other scientific progress, and it is important to do it for its own sake. So what to do? Answer, one must identify the students with high math talent and interest and give them the BEST courses possible. This doesn't mean the students
who are hard workers and great conformists--who get all A grades, but the students who conjecture and prove theorems. So, if the course is not based on theorems, but on rote procedures, you miss many such students--and turn them off.

We do this sort of selection quite well in sports and music--we don't expect most students to be talented on the piano, or wonderful football players--and we don't grouse about it. There were good programs to identify and educate young math "geniuses" in the old Soviet Union, and this was brought, for a while, to New Jersey by the Russian Mathematician ( who moved to Rutgers) Israel Gelfand. A similar program helped create the Fields Medalist Terry Tao.

I also agree that our culture ( particularly in American and other British style cultures) is antimath. That is because of the STUPID
cultural idea that intellectual abstraction is a less useful way to
understand the universe and create technology than experiment or garage puttering.

This is nonsense--in fact, ALL important technological discoveries
in the last hundred years were based on math--and often done by mathematicians.

But, the bias gets in the way of accurate history:

There is a cult of N. Telsa, because people think his work was done
by building toys in a lab--in fact, most of what people think Tesla did, was actually done by the MATHEMATICIAN Charles Proteus Steinmetz. And yet, when I was a little girl and read about Steinmetz
in a children's encyclopedia, he was shown with a screwdriver building a circuit, and not with chalk solving a differential equation.

We, in America, distrust the abstract thinker --as, at best; a useless fool, at worst a dangerous "elitist". We are a nation of
"practical people". We forget that mathematicans like Norbert Wiener
gave us cybernetics and the computer, that the mathematician
James Clerk Maxwell gave us Radio etc.

This STUPID bias gets communicated to children via science books and courses for children.

Finally, I want to point out that in General Relativity, everything that matters is in the Einstein Field Equations--and not in the philosophical bullshit ( Mach's principle, the principle of equivalence) and it is very interesting to read the ORIGINAL paper of
Albert Einstein and Marcel Grossman. This paper is in two parts--the Einstein part ( all philosophical bullshit) and the Grossman part: which introduces the use of tensors and differential geometry and a field equation which is ALMOST the Einstein Field Equation.
The Grossman Field equation doesn't have a divergence free right hand side--and the fix ( done by Einstein in his second and famous paper) is a trivial undergraduate change--already used to good effect by James Clerk Maxwell in electrodynamics--of adding a simple term to the left hand side to make the divergence of the right hand side zero.

Really, it ought to be called the Grossman Theory of Gravity--and Einstein should get " the Einstein correction term to the Grossman Field Equation".

Grossman was a mathematician.

p.p.s The only person to earn TWO PHYSICS Nobel Prizes was a Phd in Mathematics --John Bardeen.

You can of course do science without math. Aristotle used observation and deduction. A lot of scientific research today is still just experiment and qualitative observation.

(Those observations are backed by statistical analysis, which is of course math. But the legwork is where the information comes from. I wonder if the math really helps *that* much. I suspect few of the researchers themselves understand the statistics involved, except as an "is this result publishable?" oracle. Search for "most published research findings are false".)

Admittedly science's greatest hits are mathematical models that correspond closely to reality. But science is not dependent on math and I trust experiment more. More people can disprove an experimental result than can find flaws in a proof. Also my lack of faith in even very smart people's ability to apply these models correctly, even in seemingly simple cases, has been amply justified by experience (see MarkCC's last few posts for a recent example).

I find penny's argument that this is some cultural trait of Americans a trite and mildly offensive argumentum ad hominem. Skepticism is necessary to science.

Re Clive #26:

With respect to the question about a toddler:
Our brains are, at least in some facets, absolutely remarkable computers. When a child swats a fly, their brain is doing an amazing computation to figure out where the fly is (by using binocular vision), and another amazing computation to figure out how to put their hand into that position.

It might not be conscious, but it's still computation. Math is taking that intrinsic skill at certain kinds of computations, and formalizing it, allowing us to talk about it and understand it. A child may not know that their brain is doing a complex computation to figure out how to swat the fly - but we can describe how they did it using math. Seriously: how can you explain how a child figures out how to move their arm to swat a fly in a particular location, without using any math in the explanation? My position is that you can't.

With respect to your point about relativity: how could we show that information was transmitted faster than light without using any math? How can we show that transmitting information faster than light is inconsistent with relativity without using any math?

Well put. I continue to enjoy your posts, so keep them coming.

A mathematical argument is convincing to the degree that (a) you think the model is correct; (b) you think you understand the model; (c) you think the math work was done properly. In the case of MarkCC and the wind cart, all three requirements fall flat on their faces. (a) I have no real doubts about Newtonian physics, but every model MarkCC has set forth so far, to my knowledge, has been flawed in elementary ways. (b) My faith in my ability to do rigid-body analysis is shaky, but I don't have time anyway. (c) I don't think MarkCC thoroughly analyzed either of the two models he proposed, or he would have found the errors.

A YouTube video of an experiment is convincing to the degree that (a) you believe the experimenters were acting in good faith, i.e. it's not a straight-up con; (b) you understand the design of the experiment and think it sound; (c) you think unknown confounding factors are unlikely to be an issue.

MarkCC says math is absolute. Sure; but errors in applying the Newtonian model are easy to make and hard to find.

So I still don't see why we are to prefer MarkCC's revised revised math over experiment. I admit I'm more likely to use a pen and paper than a hammer and nails to look into this, but that's just because I'm lazy (and pretty good at algebra).

It's amusing that those comments regarding physics not being math fail to realize that physics is the most successful scientific discipline precisely because it is the most mathematical. The trend in any scientific discipline's progress is to increasingly become more rigorous and formalized, which is another way of saying to become more mathematical. Whether or not physics is math or vice versa is a more nuanced and profound discussion, but regardless of that outcome, whenever you are doing good science, you invariably must be doing good math. So if you want to discover the truth in the best way we know how within this world, you use science, which if done well, means doing math.

"The short version of the answer is remarkably simple: math provides a tool where you can, without ambiguity, prove that something is true or false."

Agreed. For the proof to be valid, though, the mathematical reasoning must be correct. In the previous post about "why the DDWFTTW cart works", you use math to "prove" that a rigidly-geared cart can't move faster than the belt that pushes it:

So the static look at the gear system can give you a sense of how the wind powered vehicle might work, but its rigidity limits its speed to less than the speed of the belt providing the force that's pushing it.

However, such carts can and do work. Why not post a correct analysis, instead of leaving the incorrect one up there to confuse people?

(1) Math presupposes the philosophical validity of Formal Logic (Boolean, Robbins, Wolfram, or other equivalent construction as you choose) as a basis of inference. Not very controversial, but worth noting in passing.
(2) Modern Mathematics is based on accepting the self-consistency of joint affirmation of the Zermelo-Fraenkel axioms. Again, not news to mathematicians, but worth noting.
(3) If there is to be a possibility for finitely resolvable self-consistent general rules of philosophical inference for how underlying Reality produces experienced Evidence, the rules for this production must be congruent to some unrestricted Chomsky Grammar, which may be constructed via the ZF axioms. This may be news to some parts of both the math and philosophy worlds, despite being over half a century old.
(4) This implies we experience Evidence of a universe which is Turing-Computable, as per the Effective Strong Church-Turing Universe Thesis, and at most RE-complex. This inference will be trivial to the mathematicians (and of minimal professional interest, since it deals with "reality"), and a rude shock for some philosophers.
(5) Under the assumptions of computable complexity, it has been shown (independent of the Axiom of Choice) that the description of evidence most probable correct is identified by Minimum Description Length Induction, a rigorous mathematical version of Occam's Razor. (See Minimum description length induction, Bayesianism, and Kolmogorov complexity by Vitanyi and Li [doi:10.1109/18.825807].) This may raise eyebrows for pretty much everyone who hasn't encountered it.
(6) Assuming I'm understanding the paper correctly, this means that the methodology of Science -- referring to the process of gathering evidence, forming conjectures about the evidence, developing a formal hypothesis which indicates how the current evidence may be described under the conjecture, competitive testing under this formal criterion of all candidate hypotheses (especially the body of hypotheses known as "Theory" due to previously testing best thereby) -- is the algorithm which will identify which of any group of suggested models is most probably correct based on the information of the current evidence... presupposing only the assumptions mentioned in items 1 through 3. This makes Creationists and other anti-science zealots very, very, very unhappy... but that is incidental to it being true.
(7) Using (6) requires math.
(8) Understanding these steps takes quite a bit of math. =)

Ergo: Math

The best answer to the question "Why Math?": y0

About math and physics: I have studied physics, I graduated at it, and what struck me most is the relation between math and physics. Everything physics you do, you use math. Perhaps not when *doing* an experiment, but what about setting things up, thinking through how to perform the experiment and analyzing the results afterwards? That is all math.
Math is THE language in which physics is described. To people who ask me about it, I always say that physics is nothing more (but also nothing less) than applied math. Maybe not 100% exactly always true but for simplicitie's sake it is very effective. Math is "just" formulas, physics is linking those formulas to the world around you. But it also works the other way around: when looking at the world around us, coming up with some kind of relation between things, it is math that tells us the solutions to the equation. And then with those solutions, we look back in the real world to find out what they mean.
Sometimes the math all needs to be invented, created, to solve some new equations, but often enough the solution has already been found years before physics finds a use for it. And not always just the "trivial" equations, as with the solution to the Schrodinger equations for a hydrogen atom.

That, to me, is the beauty of the "marriage" between physics and mathematics.

About this post: excellent! Mark, I really admire how you can explain why math is so beautiful, essential, intrinsic in all we think and do. Even if we don't know it, and even if we don't want to know it ;-)

Jason,
I should have said "modern physical science" not science. Of course,
things like the germal cause of diseases, and Darwin's descriptive work were done without math. Modern physical science is based on math--starting in Chemistry with the work of Dalton and later continuing with physical chemistry, and quantum chemistry.

Physics was NOTHING until the math was applied, by people like Galileo and Newton and their successors.

As to Aristotle--his "science" is a joke. He wrote such things as
"thought is based in the heart, and the brain exists to cool the blood." Luckily, when I was about eight I realised that Aristotle was a fool, and discovered Michael Faraday and Charles Darwin. Thank you, Mortimer Adler--for publishing the "Great Books of the Western World" series.

But, modern biology, including modern genetics is completely based om math--because the instruments ( such as electron microscopes, electronic measuring devices, ultramicroscopes, nuclear tagging etc.) are all based on MATH.

My comment on American anti-intellectualism is no "against the man",
argument, but the result of a lifetime of living in America--and seeing it in action.

Skepticism is healthy, but ignoring the importantce of math is not skepticism, it is idiocy!

p.s. The main reason why Aristotle is elevated is that the church --in the form of Thomas Aquinas--used him to support Christian dogma.
If you want a better ancient greek, try Empedocles or the mathematician Erathostenes.

I personally find the liberal arts curriculum, with its teaching of Aristotle and other iconic obsoletes etc., to be ill suited to prepare people to live in a modern science and math based world. More Apollonius on Conic Sections, and less Plato, please. Again thank you M. Adler for early exposure to ALL of the those greeks.

Of course, even Plato was a major supporter of math. In fact, he stated that mathematics was the fundamental basis of everything, because of his idea of ideals.

"Let nobody ignorant of geometry, enter these groves ( of academia)",
Plato(n).

Mark,
A toddler is NOT doing physics, nor is a mosquito--physics is the application of mathematical proof and rational thought and rational experiment to the deduction of the properties of matter and energy and space and time from a minimal set of basic axioms and equations.

To understand this, see Newton's Principa. Newton called it "Natural Philosophy". I like that name better because it is more self-descriptive.

Hello Mark. A mutual friend over at Rosenhouse's blog insinuated that I might like your blog.
http://scienceblogs.com/evolutionblog/2008/12/dembski_concedes_the_obvi…

Indeed, I do. So I added it to my blog links. Congratulations.

NS
http://sciencedefeated.wordpress.com/

Hi Mark,

There is this math book by William Byers,'How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics', from Princeton University Press, 2007.

Gregory Chaitin has a review quoted on a PUP webpage:
"As William Byers points out in this courageous book, mathematics today is obsessed with rigor, and this actually suppresses creativity.... Perfectly formalized ideas are dead, while ambiguous, paradoxical ideas are pregnant with possibilities and lead us in new directions: they guide us to new viewpoints, new truths.... Bravo, Professor Byers, and my compliments to Princeton University Press for publishing this book."

Re MarkCC (#31):

With respect to your point about relativity: how could we show that information was transmitted faster than light without using any math? How can we show that transmitting information faster than light is inconsistent with relativity without using any math?

Sorry - my mistake, and it was late! So having just now refreshed (somewhat) my limited and incomplete understanding of special and general relativity, I see that wasn't a good example. Something to do with the speed of light in a vacuum being constant for all observers would have been more like it, as this is an axiom for the special theory of relativity.

What I was after was something that could (hypothetically at least) be observed in a fairly direct and straightforward way (without any significant mathematics being absolutely necessary) and that would also invalidate what we could call an axiom of the theory. In other words, a theory can be invalidated not only by it generating (using mathematics) predictions that fail, but also by finding the "axioms" that is it built on are not always true.

I do realise that doing something like putting one observer on the moon and having another here (or near to) Earth would certainly involve some mathematics, but in terms of showing that we can send information faster than light (without mathematics) the actual act of a "man on the moon" directly observing that a pulse of "light" consistently arrived half a second later than the signal via the hypothetical faster than light process would not directly need any mathematics to speak of (and where both were sent simultaneously from Earth).

Re penny (#40), and knowingly running the risk of being accused of silliness, let me quote from first paragraph or two from each of the Wikipedia entries for "Physics" and "Toddler". Emphasis added by myself:

Physics (Greek: physis - ÏÏÏιÏ) is the science of matter[1] and its motion.[2] It is the science that seeks to understand very basic concepts such as force, energy, mass, and charge. More completely, it is the general analysis of nature, conducted in order to understand how the world around us and, more broadly, the universe, behaves.

Toddler is a common term for a young child who is learning to walk ... The toddler is discovering that they are a separate being from their parent and are testing their boundaries in learning the way the world around them works.

I claim this at least suggests a physicist and a toddler might have quite a bit to talk about (if they could find a common language), but I also suspect that any discussions may need to go lightly on the formal mathematics!

We can easily choose a definition for "doing physics" to force the conclusion that "you can't do physics without doing mathematics". Clearly in today's world of professional physicists, mathematics is absolutely crucial. But it seems to me to be pushing things just a little too far to suggest that you can't achieve any understanding of the universe that we live in at all without always needing to use a lot of mathematics. But perhaps that then also comes down to how you want to define mathematics! :-)

> It struck me that while there were a lot of really
> great science bloggers - people like Orac, PZ Myers,
> Tara Smith, and so on - that I didn't know of
> anyone doing the same thing with math.

Show me an easy* method of writing equations in a blog, and I'll show you a couple hundred bazillion math bloggers.

The cold fact of the matter is there isn't really any way to write math equations without using some version of LaTeX, and LaTeX is fundamentally designed to "publish on paper"... and there is no current real** alternate method that is actually suitable for digital publishing.

* by "easy", I mean, "I can write an equation without downloading and installing a bunch of modules for my blogging software and then writing LaTeX code in a blogging window", or "using MikTeX and PostScript, and converting the output file to a rasterized image which I then upload to my blog".

** by "real", I mean, "doesn't suck".

Penny wrote:

Math is NOT a language--and I am getting tired of the meme that math is a language ( clearly invented by some liberal artist).

She also wrote:

Physics was NOTHING until the math was applied, by people like Galileo and Newton and their successors.

Galileo wrote:

[The universe] cannot be read until we have learnt the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word.
Opere Il Saggiatore p. 171.

Thony,

Nice quote. Too bad that quotes have no importance in either science or math. Math is NOT a language. Statements by authority figures are not important. Mathematicians and Scientists are taught that is the first principle of Science. Liberal Artists are taught to use proof by authority and quotable quotes all the way through their "education", which is why we must abolish liberal arts education.
For example, when a liberal artist needs to know about a scientific issue he will do " 99% of scientists in this field state....". And that is a dangerous bit of trash thinking, especially when important
policy issues are considered, because science is about ideas-- not about counting votes or quoting authority.

Unlike, you were taught in your liberal arts training, that Galileo
said that math is a language, doesn't mean that math is a language.
Quotes by scientists or mathematicians mean NOTHING in science.

Besides, you are taking his comment out of context. He was discussing the idea that math describes nature. Galileo knew what math was, for he formulated and proved theorems--he was a math prof.

Math contains a language, it isn't a language. Ask any math prof.

Anyway, maybe you are showing that "math is a language" was not invented by a liberal artist but by a mathematician. We should first look for earlier sources. Francis Bacon is a good guess, I think--and it probably predates him. Anybody up for a google search?

Damn keyboard delete demon.

I meant to say " ask any math prof for a longer explanation of why math contains a language and math is not a language".

I even typed it. It just scrolled away, and I didn't notice until after it posted.

Also, I should qualify my comment on getting rid of the liberal arts. If people want to study English Literature, or History or French Poetry ---fine.

But, they should upgrade the education away from the "quotable quotes" and "proof by authority" etc., that plague the current obsolete system.

They might also require a lot more math and science. It is interesting to look at the original seven liberal arts--which did not include history or literature but those subjects such as Rhetoric
, logic, mathematical astronomy, Geometry, mathematical musical composition ( aka the math of music theory) that taught people how to think and argue rationally.
One might venture to call the seven liberal arts--the basis of the medieval university education--a SCIENCE and math curriculum.

Of course, theology was included--but, those were the times.

Better than I thought--no theology in the seven.
From Wikipedia:

* the Trivium

1. grammar
2. rhetoric
3. logic

* the Quadrivium

4. geometry
5. arithmetic
6. music
7. astronomy

Yup, a science and math oriented education! Logic, Arthmetic
( which must have included the number theory part of say Euclid)
mathematical astronomy ( such things as epicycles)....

Maybe, ( smile), we should start the "liberal arts" education in this way. Back to the seven!!!!

Pat,
what we math people seem to do generally is to write the comments in latex code, which we have learned to read. It isn't all that hard--especially for simple equations that might show up in a blog.

For example: x = \frac{-b \plusminus \sqrt{ b^{2} -4ac} }{2a}

best
Penny

Penny wrote:

As to Aristotle--his "science" is a joke. He wrote such things as
"thought is based in the heart, and the brain exists to cool the blood." Luckily, when I was about eight I realised that Aristotle was a fool, and discovered...

She also wrote:

Yup, a science and math oriented education! Logic, Arthmetic...

Now that you have found out how to use Wikipedia go away and find out who invented formal logic...

Penny wrote:

But, they should upgrade the education away from the "quotable quotes" and "proof by authority" etc., that plague the current obsolete system.

She also wrote:

Math contains a language, it isn't a language. Ask any math prof.

...!!!???!!!...

1) Time is locally homogeneous.
2) Noether's theorem.
3) Mass-energy is locally conserved.

math provides a tool where you can, without ambiguity, prove that something is true or false.

That has counterexamples. Mathematics is not empirical, merely self-consistent. String theory has 10^(10^5) acceptable vacuum solutions and rising. The Standard Model arrives massless and no Higgs is the safe bet, SUSY won't show, quantized gravitation overall is a bust. Macroeconomics is a $90 trillion failure. Interfaces have impedence.

4 - 10 = 9 - 15
Add 25/4 to both sides,
4 - 10 + 25/4 = 9 - 15 + 25/4
Write sides as complete squares,
(2 - 5/2)^2 = (3 - 5/2)^2
Take the square root of both sides
2 - 5/2 = 3 - 5/2,
add 5/2 to both sides
2=3

... and then there is Gödel

First of all, Penny ROCKS! Unfortunately, she has somewhat erred slightly in the same fashion as those she is ranting against. Yes, Penny, we do not say of the man that invented logic that he is an idiot. Come on now....

No he did not invent formal logic, though. That was something like Boole or Frege maybe. Modern formal logic is, in fact, specifically a departure from traditional Aristotelian logic.

Also, another rarely appreciated fact of history is that Plato invented math. Oh, of course there were all sorts of "mathematicians" in antiquity, but the kind of math you are talking about, Penny, is really a specific outgrowth of Platonist philosophy. It was a bunch of Platonists, like Eudoxus that almost surely inspired the whole idea of "rigor" that Euclid expresses in his Elements. So, don't be so hard on the philosophers, especially since most of what irritates you, here -- all the "math is a language" BS -- is really just an artifact of anti-philosophical empiricism.

However, I will grant you that Aristotle wasn't as great as he is often thought to be. A lot of what made him possible was really just Plato. Plato, and the Platonist geometers were some serious folks the likes of which would not be seen all in one place like that again, probably, until Gottingen.

At any rate, rock on Penny! :o)

Also, I don't mean to be too much of a weenie, but...

"No informal argument will ever produce a conclusive result."

Really? I disagree. Prove me wrong (conclusively).

Theony,
I am sure that the people of Ur, who were solving fourth degree algebraic equations and generating Pythagorean triples--thousands of years before Aristotle knew "formal logic"--for they knew how to prove theorems.

Of course, the liberal arts education--glorifying Aristotle claims that he invented formal logic--and that is pure baloney. In the same way, they claim that the Greeks invented proof in Math--also baloney.

Thomas Aquinas strikes again.

Adrian,
I too learned the myth that Aristotle invented formal logic, and that Plato invented math--from college books of my sister when I was around eight--just after I was reading Adler's wonderful series.
But this is NONSENSE--and liberal arts classes still teach this NONSENSE.

Neither Plato nor Aristotle invented rigor in math or mathematics--Thales of M, who died in 547 BCE ( Aristotle was born in 384 BCE and Plato was born in 428 BCE) already had proved theorems on similar triangles and the theorem ( known as Thales Theorem) that the base angles of an isoceles triangle are equal--later adopted as the
"bridge of Asses" by math loving Plato.

But, all this really predates the Greeks:

It is true that we have sources for the Greek textbook approach to math--and that we call that "rigorous math" in our older histories. But, How can one discover the principle of generation of Pythagorean
Triples--using what is essentially a uniformization of the circle--without a concept of proof? This was already known in Ur, thousands of years earlier.

Much of "Greek" math can be traced back to India and to other earlier cultures such as Egypt and Ur. The Egyptians knew continued fractions, regularly used binary arithmetic etc. They argued with logic. But, we have few sources from these cultures.

As it happens, we have more sources from the Greeks, largely via Roman translations. So, we claim they invented rigor.

Adrian rocks too!

I agree with both of your points--modern formal logic was NOT invented by Aristotle and is the direct result of the work of people like Frege,
Russel, etc. Symbolic logic goes back to Leibnitz and then Boole.
What you say about a reaction to anti-philosophical empiricism" is a deep insight and, on reflection, I agree.

Pythagoras died 490 BCE. Plato born 428 BCE.
So Plato didn't invent rigorous math.
Nor did Aristotle, who was born in 384BCE.

The Pythgorean Theorem is a pretty sophisticated example of formal logic--and it, actually predates the Greeks. But, the above dates tell us it predates Plato and Aristotle.

Are you sure? The Pythagorean Theorem is as attributable to Pythagoras as Fermat's Last Theorem is to Fermat. (In fact, Fermat's Theorem is probably more attributable to Fermat.) When was the first rigorous proof? How about the first axiomatic development? I don't think anyone even cared about such things until Plato. It is all an outgrowth of the idea that knowledge consists of more than merely true belief. The call for rigor in mathematics is precisely a result of emphasizing *justification* as being equally important to (if not more important than) just being right. And, that really is Plato, not the Indians or the Egyptians. It's not even "the Greeks" or something like that. It is really just that one guy who happened to be Greek and his legacy. You can see this build up from informal Socratic dialogs in which people that seem authoritative or common sense ideas all seem to fail to withstand scrutiny. This passes into serious and extended attacks on philosophical skepticism and defenses of objective knowledge with a focus on the justification for beliefs. And then, finally you have Aristotle fleshing out essential aspects of legitimate justification (e.g. the laws of contradiction and excluded middle).

The *rigorous* development of the method of exhaustion by Eudoxus -- the true basis of calculus -- was just another step in the process. And Euclid putting all of this and more together in a treatise on Greek mathematics was little more than a butterfly collection at that point. But, before Plato, no one would have even cared -- about ANY of it. They would just "see that the theorem is true and move on", like most calculus students do today....

At any rate, I'm glad to see someone defend math for a change. Everywhere I go, I rarely see it -- even most mathematicians completely give into the status quo where the best examples of math are all physics problems not math problems and actually teaching a kid to prove theorems constitutes some kind of child abuse. I am certain they will soon send CPS out to collect my kids because I teach them math out of some old Frank Allen texts from the New Math era.

Adrian,
Yes, I am sure. The Pythagorean Theorem was certainly proved by the time of Pythagoras in Greece--and the question is which member of his
study group ( called his cult by our writers) proved it. They also
proved that the square root of two is irrational--which predates Plato. This is a fairly sophisticated number theory proof.

From wikipedia:
//The earliest known use of irrational numbers was in the Indian Sulba Sutras composed between 800-500 BC.[citation needed] The first existence proofs of irrational numbers is usually attributed to Pythagoras, more specifically to the Pythagorean Hippasus of Metapontum, who produced a (most likely geometrical) proof of the irrationality of the square root of 2.
//
//Hippasus of Metapontum (Greek: ÎÏÏαÏοÏ), b. c. 500 B.C. in Magna Graecia, was a Greek philosopher. He was a disciple of Pythagoras. To Hippasus (or Hippasos) is attributed the discovery of the existence of irrational numbers. More specifically, he is credited with the discovery that the square root of 2 is irrational.
//

There is no real proof though that Hippasus was murdered for his discovery, which was a story we were taught in school.

Plato did NOT invent or inspire the idea of rigorous math--put that to rest:

In fact, in one of Plato's famous dialogs on the nature of the Good, Socrates interviews a famous research mathematician--who had found a unification of the proofs of the irrationality of many integers--essentially, the rational roots theorem, and ask him what is the nature of the good, to show that math is easier than other branches of philosophy.
The mathematician starts enumerating the kinds of good--which Plato uses to show the greater difficulty of the question.

Although it is unclear that Socrates actually said the things attributed to him by his student Plato--it is clear that the audience knew this mathematician and his accomplishments.

Crap taught to us in school dies hard. Most of the history we have learned in school is crap--and this is especially true of the history of science and math.

Adrian,
The new math was a great success. It was created to turn on future math geniuses--who were turned off by rote arithmetic--and to thus help us win the cold war.

It was so successful, that by the time the new math generation left graduate school, there was a huge glut on the job market for USA research mathematicians. The new math generation ( my generation) made
tremendous success in research mathematics--America was the best research math center in the world, for quite a while.

In fact, some of my earlier comments on the teaching of math were based on my experiences with the new math--I was one of the students in experimental classes where it was developed.

It was NOT created to teach average Johnny how to add quickly. In that, it was a dismal failure--and that was considered ok, as the creators knew computers would be everywhere in a few decades.

So kudos you Adrian, for using new math texts in your teaching.

Adrian,
Note that Thales had gone to Egypt to study geometry, and that he gave
proofs of geometrical theorems--such as Thales Theorem. This was cited by many later Greeks-notably by Aristotle, who called Thales the originator of Greek Philosophy ( in the science and math sense). Essentially all of the correct science in Aristotle is actually quoted
Thales.

The only question is whether Thales proofs were his or whether they
were already well known in Egypt. Certainly, the Egyptians knew such things as the use of tangent a thousand years before Thales. But, Thales wrote proofs. We don't have his originals ( so much of ancient Greek writing is gone), but we do have the discussions of his proofs by other Greeks.

Thales is long before Plato, and he contributed to Math, to science and to astronomy. This includes computational astronomy--the prediction of eclipses and the revision of calenders.

When I taught my first university level course in the history of math--at UCLA, I learned that most of the stuff I had learned in school about the history of math was obsolete. It was an eye opener.

Adrian,
Just as an aside, most of the "ideas" in Plato are already in
Parmenides. He was considered the Originator of the other kind
of Greek philosophy ( not science and math).

My favorite Greek (non science and math) philosophers are the Stoics--who have gotten me through tough times. Of course, their ideas
were known in India thousands of years earlier--just like the ideas of
Parmenides.

Rock on Adrian.

The emphasis on applications by math profs is a moral failure. It is due to the pressures of what I called " the anti-intellectual American
...".

Although, some of my own research is in mathematical physics, I have fought against this trend for decades.

It starts in elementary school with the widely touted idea that "applications make math relevant and easier to understand". That is very popular with the NON-Mathematicians who end up teaching elementary math. They tend to be very concrete thinkers.

However, it has been my experience that applications muddy the waters and make math harder to learn. Applications are best learned AFTER the math.

Let me add a comment by, and a comment about, the relationship between Math and Physics by the most famous Mathematical Physicist that the public never hears about, as they have had decided for them that Steve Hawking is the only game in town.

As mentioned in his St. Andrew's University Mathematical Biography, speaking at the American Mathematical Society Centennial Symposium in 1988, Edward Witten explained the relation between geometry and theoretical physics:-

It used to be that when one thought of geometry in physics, one thought chiefly of classical physics - and in particular general relativity - rather than quantum physics. ... Of course, quantum physics had from the beginning a marked influence in many areas of mathematics - functional analysis and representation theory, to mention just two. ... Several important influences have brought about a change in this situation. One of the principal influences was the recognition - clearly established by the middle 1970s - of the central role of nonabelian gauge theory in elementary particle physics. The other main influence came from the emerging study of supersymmetry and string theory.

Basically Witten is a mathematical physicist and he has a wealth of important publications which are properly in physics. However, as Atiyah writes:

Although he is definitely a physicist (as his list of publications clearly shows) his command of mathematics is rivalled by few mathematicians, and his ability to interpret physical ideas in mathematical form is quite unique. Time and again he has surprised the mathematical community by his brilliant application of physical insight leading to new and deep mathematical theorems.

Atiyah adds, expressing the same ideas in the following way:-

... he has made a profound impact on contemporary mathematics. In his hands physics is once again providing a rich source of inspiration and insight in mathematics. Of course physical insight does not always lead to immediately rigorous mathematical proofs but it frequently leads one in the right direction, and technically correct proofs can then hopefully be found. This is the case with Witten's work. So far the insight has never let him down and rigorous proofs, of the standard we mathematicians rightly expect, have always been forthcoming.

By the way, he would either laugh or be annoyed (I don't yet know which) to have it pointed out to him what I just discovered a few minutes ago: The prime 2681951 denotes the birthday, 26 Aug 1951, in Baltimore, Maryland, of Edward Witten, Fields Medal winner in 1990.

Say, that's an unusual thing: a Physicist winning the Field Medal. It's almost as if Physics and Mathematics were related...

Jonathan,
Excellent. Witten gets lots of public press here in America. He is widely called...." the next Einstein" in the press and on TV. He's pretty good-but not THAT good, yet. His seminar and conference lectures are very inspiring, though most mathematicians who attend his talks feel like they are drinking from a fire hose of information!! I know I do, and it is a great feeling.

The correct proofs of deep conjectures by Witten were done by others, actual mathematicians--who are obscure to the general public. And, Witten shared his
well-deserved Fields Medal with a far more publically obscure
mathematician. Witten helped create topological Quantum Field Theory, and several other branches of physically motivated topology. He has made unifications and simplifications in knot theory--which require quite deep insight.

Witten and his student Seiberg, also pioneered the Seiberg-Witten equations ( motivated by Supergravity) that have simplified many proofs that were very difficult using Yang-Mills Theory.

There is, of course, a relation between mathematical physics and mathematics.

Historically, physics has inspired a lot of math research--such fields as celestial mechanics, electrodynamics, general relativity, and classical quantum mechanics were major inspirations for math.
Witten has shown us the great inspiration of quantum field theory.
He's amazing!

Here is a list of important physicists:
Galileo, Newton, Euler, Lagrange, Lord Rayleigh, Kelvin, Carnot, Maxwell,
Dirac, Born, Bardeen, Hawking, Penrose.

Now go back and read it again, noting that these people were all math professors--with the exception of Born--who had a Phd in math, and was Hilbert's first research assistant.

So this post is titled: The influence of math on physics.

As I said, in my first post, math and physics are deeply connected,
remove the math from physics and nothing worthwhile is left.

I tend to agree with this Wiki page Mathematics as a language.

I particularly like the points that sign languages are recognized and the quote from Ford and Peat (1988).

Mathematics is certainly Communication.

Doug,
Communication of what?
It is the CONTENT of math--its concepts, its theorems, its proofs that is the part of math that is NOT a language.
Math contains a language for communicating and expressing these concepts.

Three points on my initial reaction to Penny's comments and Shahn Majid's philosophy of physics, hotlinked below.

ON THE NATURE OF PHYSICS

(1) Mathematics, a sort of shorthand, is rich in ambiguity because of its brevity.

Mathematics is both a language (signs, rules for combining signs, rules for precedence of one sign over another, etcetera), and part of the content of that language (the mathematical ideas that can be expressed in the symbolism). What Chemists are interested in are the Chemical concepts that can be expressed in Mathematics. What
Biologists are interested in are the Biological concepts that can be expressed in Mathematics. What Experimental Physicists are interested in are the structures and patterns and changes in the experimental data that can be expressed in Mathematics.

And, as S. Magid put it: "the goal of theoretical physics, at least in my opinion, is to arrive at our best understanding of the fundamental processes of nature. We use of course the language of mathematics, and most physicists (including me) would ideally like to find a single elegant theory powerful enough in principle to explain all known phenomena. In practice it means our existing domains of understanding should be recovered as special limits."

(2) The '=' sign has at least four mutually exclusive meanings; identity, equality, implication and correlation." I've seen different shades of these meaning used in Geometry, where bad textbooks and bad teachers confuse equality and congruence. I've seen the sign used in
Mathematical Logic and in the Electronic Engineering that embodies Boolean Algebra to mean a specific relationship between input and output.
Failure to specify which means the equation is meaningless. Or at least depends on interpretation, and thus can lead to useless argument.

(3) Mathematics which has taken over a physical subject, such as Electromagnetic Theory, or String Theory, and gone where the Math is pretty but the experiments not in line with prediction, is deeply flawed."

Magid: "The starting point is that Nature after all does not know or care what mathematics is already in textbooks. Therefore the quest for the ultimate theory may well entail, probably does entail, inventing entirely new mathematics in the process."

Americans can find on the newsstand right now the so-called January 2009 "Discover" magazine, a special issue on "100 Top Science Stories of 2008." Interesting summaries and pictures of paradigms collapsing in some cases, and consolidating in others. There is also a spirited defense by his daughter of the late great Fritz Zwicky, inventor of Dark Matter, gravitational lensing, neutron stars (partly), supernovas (partly), carpooling (sort of). What's of extra interest to me, who'd met and spoken with him, is how he was defamed, censored, marginalized by the Establishment figures he ridiculed. Including the late Jesse Greenstein, who was my Astronomy Mentor.

I've commented in this blog a number of times about Fritz Zwicky's theory of the Ideocosm, the space of all possible ideas. Magid makes an interesting philosophical question about the Mathematical portion of the Ideocosm:

"'in the tableau of all mathematical concepts past present and future, is there some constrained surface or subset which is called physics?' Is there an equation for physics itself as a subset of mathematics?"

Experts can differ on this. Is Math part of Physics? Is Physics part of Math? Are they two different approaches, practiced by two different communities? These are what I've called differing metaphysical stances.

In an ideal world of rational people, which does not exist, special attention would be paid to iconoclasts who make many correct predictions: Alfven, Tommy Gold, Zwicky. But the world is very very far from ideal, and these folks are mercilessly attacked. It takes decades for anyone in the Establishment to ask: was there something correct in what these people said and did?

Meanwhile, we can be thankful that Mark Chu-Carroll keeps such essential discussions on the front burner.

MarkCC: determinants of mod-12 matrices, without realising it? Srsly? Are you telling us that they supplied your brother with some sort of addition and multiplication tables for Z/12Z encoded into musical notes, and he had to do determinant computations using those, or what?

It seems perfectly plausible to me that some of the things done in a serial composition class would be related to arithmetic mod 12. But determinants? That's really hard to swallow.

To those who think that "mathematics is a language": OK, so what sort of language feature does (say) the concept of an abelian group correspond to? How about Wiles's proof of Fermat's last theorem? You can, in principle, describe a language fairly completely in one (albeit very large and formidable) book; do you imagine that you could describe mathematics completely in anything less than, say, a thousand books? (I think the correct number is considerably more, for what it's worth.) You can use (a little bit of) mathematics as a language; you do that any time you write something algebraically, for instance; but what makes that worthwhile is precisely the fact that there's a lot more to mathematics than a handy way of expressing things.

Well, Plato, Parmenides and the Stoics aside, one thing I think that is not coming explicitly out in these discussions about the relation of math and physics is that when someone says "math is a language", they may not realize that they are really *limiting* the field. What Penny is saying is that there is something else to it besides just a "language". Ironically, most of the people that look at math that way tend not to be mathematicians (though I have definitely heard even mathematicians say stuff like that). In fact, because of our K-14 educational system most people don't even experience math outside of some sort of an empirical science of calculating values or plotting points, perhaps plugging it into some computer model.

No one is saying that these subjects (computing, theoretical physics or whatever) don't exist. We're just saying there is another subject, as well. If this other subject doesn't go in math, I wouldn't know where you would place it -- philosophy, I guess (which would definitely upset Penny). It isn't anything like these empirical sciences and it's value isn't because "it works" (although it does work). It's just a different subject matter. That is all.

"...do you imagine that you could describe mathematics completely in anything less than, say, a thousand books?..."

Didn't the Bourbakis do that?? I'm pretty sure it was less than a thousand. :o)

In fact, come to think of it, I thought it was supposedly all contained in e^(i*pi)=-1 or something. (That's what I learned in physics at any rate....)

MarkCC: determinants of mod-12 matrices, without realising it? Srsly? Are you telling us that they supplied your brother with some sort of addition and multiplication tables for Z/12Z encoded into musical notes, and he had to do determinant computations using those, or what?

It seems perfectly plausible to me that some of the things done in a serial composition class would be related to arithmetic mod 12. But determinants? That's really hard to swallow.

Yes, I'm absolutely serious about the mod-12 stuff.

I don't remember the precise details (it's been about 22 years since this event!), but it made a real impression on me, so I still remember the basic idea. Basically, they would assemble a matrix from the intervals between different notes in a sequence. So a sequence step from C to D# would be a minor third - aka 3m. Then they had rules for how to combine intervals - which were, essentially, a funky multiplication table. The rows of matrix was the interval separations for different lines in the score.

I don't remember why they did this; it was something to do with finding certain elements of the basic tone-row in the piece, but it must have been more complex, because just recognizing the tone row is easy once you've converted to the interval notation. It must have been some way of discovering relations between different score-lines, but damned if I know what they were. (My music theory frankly sucks; I know some basics, but that's all. I'd love to improve it, but I've never been able to find a decent approachable text. If anyone has any recommendations, I'd love to hear them!)

Adrian,
Exactly. You have made my comment about math being more than a language quite clear.

Bourbaki was a collection of graduate level texts on math--at the time it was written--and it covered those things that a second year graduate student should have known. On the other hand, a lot more math has been discovered since then.

But, yes, a thousand books could be extreme--probably a few hundred would cover most of the key ideas in current mathematics.

The comment on e^{i \pi} was clearly a joke by your physics teacher.

Jonathan,
I usually don't read Discover Magazine, but I did read that article
yesterday, in a Starbucks waiting for a friend. It was a very interesting article.
It was interesting that no mention was made of the work of Oppenheimer and Snyder on collapsed matter.

In re: tired light. Ok, that was wrong, but one should look up
Emile Wolf's paper on the "Wolf shift", for a similar--correct--and disregarded spectral shift, that could have important astrophysical implications. It is proved by the mathematics of electromagnetic theory.

Actually, Oppenheimer-Volkoff.

Of course, the theoretical work of Oppenheimer and Volkoff was inspired by the basic idea of Zwicky and Lev Landau.
Zwicky was clearly a major creative force--and the article is very interesting.

> what we math people seem to do generally is to
> write the comments in latex code

Whelp, I'm a "math people", and I don't write comments in latex code.

> It isn't all that hard--especially for simple
> equations that might show up in a blog.

Sure. That's not the point. Morse code is really easy to read too, if you take the time out to learn it.

Taking your example... "x = \frac{-b \plusminus \sqrt{ b^{2} -4ac} }{2a}"... look at it from an efficiency standpoint. LaTeX is designed as a typesetting language. I can write that equation out on paper much quicker than I can type it, and I can type pretty damn fast. It's also not readily parsed by anyone who doesn't know the markup, which is the vast majority.

I want software that makes it easier for me to publish digitally, not harder. I was ecstatic as an undergraduate when I learned how easily I could shorthand, and now in order to implement that shorthand I have to encode the shorthand... in longer than longhand.

Sure, LaTeX is the best thing we've got, currently. It does the job. I'm not saying there is anything better, but it's also nearly 30 freaking years old and it predates any other markup language currently in normal use (unless you're going to argue that XML is really just SGML which is really just GML).

It's time for someone to base mathematical publishing on a tool that starts somewhere beyond the ASCII character set.

Pat,
Latex isn't thirty years old--Tex is. Anyway, maybe you should write the program you want. It's not that hard to do.

I didn't find Latex hard to learn. It takes a few days to read through
a Latex book--and learn the language. Of course, if one tries to just
learn the language by copying bits of other people's code--a common
learning strategy in the younger generation, it could take a long time
to really master the language--as you are decrypting the Mayan Codex.

Most of my math coauthors and friends really do use Latex code in our
emails--try it. It isn't so hard to do that.

On the other hand, instead--you could get really worked up emotionally, write the program you want--and get rich and famous!!
Frustration is sometimes useful--it tells you what need YOU can fill!

Pat,
You might want to take a look at some of the interfaces for Latex, some people like Lyx.

My own tendency is to use as few layers of software as possible,
to avoid bugs due to layers of incompetence.

There is even a WISYWOG interface that runs in ( horrors) Windows--called Scientific Word. It was written by a student of J.J.
Kohn.

I don't know if they have email embedding, but you could probably add it--or ask the developers to do it.

Pat,
Also take a look at the Kyle editor. Some people like it, because it's still Latex, but a lot of helpful interface features are in that editor--such as automatic environment closing and easy back and forth editing from the Latex file to the editor. Like the vim editor, it highlights unmatching parentheses.

My personal preference is just to use emacs to write Latex code--when typing a paper, anyway.

WYSIWYG, of course.

Spelling doesn't work before breakfast and coffee.

Along the lines of Pat's comment, the time may be ripe for a handwriting recognition interface for a digital writing tablet that
produces Latex code, and thence embedded postscript graphics.

Handwriting rec was used, too primitively, in early digital writing tablets--the time wasn't ripe.

Penny, quite possibly a few hundred books could cover "most of the key ideas". On the other hand, if you actually wanted all of currently-known-mathematics in those books, you'd need a lot more than a few hundred. Even if they were as terse as Bourbaki's. (Glancing quickly over my maths books: I reckon I have ~ 250; perhaps there's a 50% efficiency loss from duplicated material and another 50% from not being maximally terse -- not, of course that mathematics books should really be maximally terse given that human beings have to read them -- so perhaps the material represented on those shelves would take 60 maximally-efficient books to cover. I'd be *astonished* if that material included more than, say, 3% of all the mathematics that is known; the figure is probably much, much less. At 3%, that would suggest a figure of 2k books for the whole thing.)

The real difficulty in writing mathematics on a computer is, I think, not the ASCII character set but the use of a keyboard. I believe there already is something that runs on tablet PCs (remember those?) that does a good job of understanding hand-written mathematics.

I wonder, also, how inefficient LaTeX really is. I just did a very crude experiment, and copied some simple formulas from a number theory book (1) onto paper and (2) into LaTeX form. #2 was indeed slower, by about a factor of 3/2, but that's not a fair comparison: #2 is how long it actually took to get the formulas into the correct electronic form, whereas #1 is an upper bound on how efficient a writing-based method could possibly be. Any mistakes or glitches could inflate the time drastically. Of course, #2 also falls off a cliff when you hit stuff you don't know how to write in LaTeX and have to start checking the documentation; but the original question was "why aren't there more good mathematicians blogging about mathematics?", and I'd have thought that most good mathematicians already know how to express in LaTeX most of the mathematics they are likely to want to blog about.

G,
I didn't find Bourbaki all that terse, when I read many of them as a student. They are no more terse than say, Spanier or Federer. In fact, I tend to prefer a short terse book to a long expanded one, I am less intimidated by carrying a short book around to read. The idea of fifty 500 page books yet unread preys on me.

But, ok, a few thousand might be a better estimate, if we want more expanded text. The main point is that math is a far larger subject than the general public knows. And, if we include Journal articles---only important for researchers--it is even larger. So, in that case, it is certainly thousands of books--in fact, ten thousand or more.

Many math books are now available on Kimble etc,--and I have a new hobby of replacing my physical math library with digital versions. Journal articles are available online--for those lucky enough to have
university access or those rich enough to pay horrible fees. These
things should be available online to everyone, and cheaply or even free.

Why should some publishing company get fifty dollars for an article that they basically typeset or published a Latex file of, for which the author was never paid ( and often had to pay a publication fee), and for which in many cases--nobody in the publishing house currently had any connection to.

People scream about free access to Movies and Pop music online--this is the same issue, but FAR more important to the progress of science.

My suggestion for the future is to add a Wiki to arxiv--like wikipedia--where people discuss the papers, and correct them. I have been saying this for years. Wikipedia works, so would this.

Then, we could get rid of the publishing companies.

Mark (and everyone else who enjoys this sort of thing): check out the linked blog in #41 ("sciencedefeated") - some serious crankage there.

Show me an easy* method of writing equations in a blog, and I'll show you a couple hundred bazillion math bloggers.

The cold fact of the matter is there isn't really any way to write math equations without using some version of LaTeX, and LaTeX is fundamentally designed to "publish on paper"... and there is no current real** alternate method that is actually suitable for digital publishing.

* by "easy", I mean, "I can write an equation without downloading and installing a bunch of modules for my blogging software and then writing LaTeX code in a blogging window", or "using MikTeX and PostScript, and converting the output file to a rasterized image which I then upload to my blog".

Go to wordpress.com or wordpress.org. Sign up for the basic deal with no bells or whistles. The normal input window accepts LaTeX. I'm not sure that there is a great shortage of maths bloggers. I don't know of any other subject that has it's equivalent of Terry Tao, Tim Gowers and John Baez blogging regularly and enthusiastically. In my small way I am also a maths blogger.

Mark (and everyone else who enjoys this sort of thing): check out the linked blog in #41 ("sciencedefeated") - some serious crankage there.

Or it's the most subtle humour site on the internet. I have a discussion going on it here

You are right. Math is awesome. But you are not right with your conclusions, what means your observation is wrong. Math is just an abstract model of the real world. If one point is missing the whole model is false (or at least a part of it).
So I can proof you wrong. The human body is better than a perpetuum mobile machine. There is no linear dependence beteween food (energy) and life. So there is more than the facts you observe. So to speak there is some fact missing in your equations. I prefer to look and read what they (steorn) say until it is proven wrong - this is evolution. I do not prefer the blind, dictorial speak of an - i have to knee down and kiss his hand - academic person who just knows math and has forgotten that it is just a simple model of the real world. A model which does not work in real world. In the real world 1+1 is never 2. You have to make major major cutbacks to make this work (have you ever seen two similar apples - space, time, dna, color, size etc. they are so terrible different).
So I believe or better hope for FREE, CLEAN energy. Yes I do.

The impetus behind General Relativity, a beautiful bit of math, came from observation. And contrary observations could disprove it. The math would still be right, though.

I know it's bad form to post without skimming the comments, but this is a long thread and the quote above is just WRONG. Einstein was pretty clear that he arrived at relativity through deductive reasoning and mathematics. Yes, he provided a few physical observations that supported it (Michelson-Morley, orbit of Mercury) but these were almost an afterthought to the theory itself.

Relativity comes from first principles on how time and space should be (or even can be) measured. It was only a few decades later that our ability to measure time and space became good enough to really corroborate the theory. If someone wants to make the argument that inductive reasoning trumps mathematics in science, arguing from relativity is not the way to go about it.

I am an accidental math teacher, by fate and not design. Stumbled into it, via baseball and jazz composition... engineering? dead end. computer programming? interesting, but I am not OCD enough... but i could do the maths for my friends in college.

so here I am, a literate, musical, math teacher. The only reason they keep hiring me every year, is I can relate GCF's to shopping for clothes, composition of functions to washers, driers, and tur-duck-ens, statistical hypothesis errors to politics (conservative vs. liberal) etc., etc.

I tell them of my Quixotian quests for the General Polynomial Theorem (mostly found) and the General Probability Distribution for Dice Throws, the (Wry Mouth) Matrix -- an improvement on Pascal's Triangle, etc. students like it when the teachers are in pursiut of knowledge too.

and starting every class with some odd snippet of music -- from Bach to Gogol Bordello to Jonny Lang -- seems to work too.

As for your brother? Certainly he knows that in the dark ages, when people were stupid, music was part of the college math curriculum, no? ;o/

"The sciences do not try to explain, they hardly even try to interpret, they mainly make models. By a model is meant a mathematical construct which, with the addition of certain verbal interpretations, describes observed phenomena. The justification of such a mathematical construct is solely and precisely that it is expected to work."
John von Neumann

"As far as the laws of mathematics refer to reality, they are not certain, as far as they are certain, they do not refer to reality."
Albert Einstein

The quotations in #93 from John von Neumann and Albert Einstein are favorites. But one must be careful in interpreting them.

"By a model is meant a mathematical construct which, with the addition of certain verbal interpretations, describes observed phenomena" is valid, but it does NOT mean that all mathematical constructs, with or without "verbal interpretations" are descriptions of "observed phenomena" (i.e. physical, chemical, biological, social, economic, geological, astronomical...).

"Pure Mathematics" has constructs which do NOT necessarily have anything to do with "observed phenomena" except in the Platonic sense of observed within the formal structures of Pure Mathematics itself.

Indeed, there is a class of mathematical constructs in between the "observed phenomena" and the Pure Mathematics. These are counterfactual constructs.

IF Newton's Universal Law of Gravitation had force change with distance according to some other formula than inversely proportional to the distance between two objects, how would dynamics of moving bodies work? The Math is consistent; it just doesn't apply to our universe, although some who don't like Fritz Zwicky's "dark matter" would prefer a proportionality to distance to a fractional power not equal to 2.

IF the speed of light were 100 miles per hour, what would happen to me as I bicycle to work?

If the Second Law of Thermodynamics ran backwards, what would happen? That's the key to my Caltech classmate David Brin's novel "The Practice Effect."

It is part of the intersection of Math and Science to intentionally deviate from "observed phenomena" and to consider models which are perurbations or deformations of those models which are descriptions of "observed phenomena."

Sometimes Pure Math turns out to describe "observed phenomena." Sometimes not. Sometimes we don't know yet.

Saint Albert was critiquing Pure Math with his "as far as they [Pure Math constructs] are certain, they do not refer to reality."

The "reality" of the scientist and the Mathematician are not exactly the same reality.

@ penny

> Along the lines of Pat's comment, the time may be
> ripe for a handwriting recognition interface for
> a digital writing tablet that produces Latex code,
> and thence embedded postscript graphics.

I might cheerfully agree to assassinate someone at the behest of anyone who would successfully tackle this project.

I've looked at Scientific Word, and I actually think it's a somewhat usable program, but I don't like platform-dependent applications for critical tasks.

But enough hijacking this thread to complain about digital publishing. :)

Someone above wrote:

Statements of the form "___ is impossible." strike me as deeply unscientific and dogmatic.

I agree. Reminds me of Arthur C. Clarke's First Law: "When a distinguished but elderly scientist states that something is possible, he is almost certainly right. When he states that something is impossible, he is very probably wrong." :)

"...once you bring math into the discussion, you can force a conclusion."

Take it from someone who has spent years debating ufologists, cryptozoologists, and religious extremists: the power of a logical conclusion is directly proportional to the vehemence of a kook's insistence that it isn't so. Being able to prove a point rationally, raises the defences of anyone who needs their world to be otherwise. The better your argument, the bigger their resistance.

Just a sad aside: Israel Gelfand died yesterday.