The Unreasonable Effectiveness of Mathematics in the Natural Sciences

Chad Orzel provides the pointer to Helge Kraghe, who writes in Physics Web http://physicsweb.org/articles/world/13/12/8 about how quantum theory existed in the equations of physics half a decade before the human brain of any physicist understood it:

It was 100 years ago when Max Planck published a paper that gave birth to quantum mechanics - or so the story goes.... According to the standard story... quantum theory emerged when it was realized that classical physics predicts an energy distribution for black-body radiation that disagrees violently with that found experimentally. In the late 1890s, so the story continues, the German physicist Wilhelm Wien developed an expression that corresponded reasonably well with experiment - but had no theoretical foundation. When Lord Rayleigh and James Jeans then analysed black-body radiation from the perspective of classical physics, the resulting spectrum differed drastically from both experiment and the Wien law. Faced with this grave anomaly, Max Planck looked for a solution, during the course of which he was forced to introduce the notion of "energy quanta". With the quantum hypothesis, a perfect match between theory and experiment was obtained. Voila! Quantum theory was born.

The story is a myth, closer to a fairytale than to historical truth...

The study of black-body radiation had begun in 1859, when Robert Kirchhoff, Planck's predecessor as professor of physics in Berlin, argued that such radiation was of a fundamental nature.... [I]n 1896... Wien found a radiation law that was in convincing agreement with the precise measurements being performed at the Physikalisch-Technische Reichsanstalt in Berlin... the spectral density, u, - the radiation energy density per unit frequency - depended on the frequency, f, and temperature, T.... Planck was... interested in... establishing a rigorous derivation of it.... To secure a more fundamental derivation he... reinterpreted Boltzmann's theory in his own non-probabilistic way. It was during this period that he stated for the first time what has since become known as the "Boltzmann equation" S = k log W, which relates the entropy, S, to the molecular disorder, W.

To find W, Planck had to be able to count the number of ways a given energy can be distributed among a set of oscillators. It was in order to find this counting procedure that Planck, inspired by Boltzmann, introduced what he called "energy elements", namely the assumption that the total energy of the black-body oscillators, E, is divided into finite portions of energy, epsilon, via a process known as "quantization". In his seminal paper published in late 1900 and presented to the German Physical Society on 14 December... Planck regarded the energy "as made up of a completely determinate number of finite equal parts, and for this purpose I use the constant of nature h = 6.55 x 10-27 (erg sec)"....

Quantum theory was born. Or was it? Surely Planck's constant had appeared, with the same symbol and roughly the same value as used today. But... [Planck] explained in a letter written in 1931, the introduction of energy quanta in 1900 was "a purely formal assumption and I really did not give it much thought except that no matter what the cost, I must bring about a positive result."... Far more interesting [to Max Planck] than the quantum discontinuity (whatever it meant) was the impressive accuracy of the new radiation law and the constants of nature that appeared in it.

If a revolution occurred in physics in December 1900, nobody seemed to notice it.... Very few physicists expressed any interest in the justification of Planck's formula, and during the first few years of the 20th century no one considered his results to conflict with the foundations of classical physics.... As to the quantum discontinuity - the crucial feature that the energy does not vary continuously, but in "jumps" - [Planck] believed for a long time that it was a kind of mathematical hypothesis, an artefact that did not refer to real energy exchanges between matter and radiation....

[N]owhere in his papers of 1900 and 1901 did Planck clearly write that the energy of a single oscillator can only attain discrete energies.... If this is what he meant, why didn't he say so? And if he realized that he had introduced energy quantization - a strange, non-classical concept - why did he remain silent for more than four years?...

[I]t was Einstein who first recognized the essence of quantum theory. Einstein's remarkable contributions to the early phase of quantum theory are well known and beyond dispute. Most famous is his 1905 theory of light quanta (or photons), but he also made important contributions in 1907... Einstein's 1907 theory of specific heats was an important element in the process that established quantum theory as a major field of physics. The changed status of quantum theory was recognized institutionally with the first Solvay conference of 1911, on "radiation theory and the quanta", an event that heralded the take-off phase of quantum theory...

Max Planck comes up with an equation that works. In order to do so he has to make a "purely formal assumption." And it is only half a decade later that Einstein realizes that the little h that appears in Max Planck's equation is not a formal assumption or an "artefact" but instead tells us what is perhaps the most important thing about the guts of the universe.

For half a decade the first equation of quantum theory was there. But nobody knew how to read it.

It is this "what if we took this equation seriously?" factor that is, to my mind at least, the spookiest thing about the unreasonable effectiveness of mathematics in physics. Take the h in Max Planck's equation seriously, and you have the quantum principle--something that was not in Planck's brain when he wrote the equation down. Take seriously the symmetry in Maxwell's equations between the force generated when you move a magnet near a wire and the force and the force generated when you move a wire near a magnet, and you have Special Relativity--something that was not in Maxwell's brain when he wrote down the equation. Take Newton's gravitational force law's equivalence between inertial and gravitational mass seriously and you have General Relativity--something never in Newton's mind. And take the mathematical pathology at r = 2M in the Schwarzchild metric for the space-time metric around a point mass seriously, and you have black holes and event horizons.

Comments

[Kuas]

Another example: Murray Gell-Man's introduction of "quarks", based on his use of SU(3) representation theory, which he said very explicitly at the time to say were not to be thought of as real particles but mere mathematical abstractions. He has since tried to rewrite history on this issue.

[Kaleberg]

Dirac was more positive. In 1928, he predicted the positron based on the fact that the sign of the charge in one of his solutions was indeterminate. That meant there had to be both positive and negative electrons. The positron was discovered in 1933.

[Anderson]

"Never trust the teller, trust the tale." --D.H. Lawrence

[anne]

http://www.nytimes.com/2005/04/08/opinion/08greene.html?ei=5070&en=0d30b3c8c6750ba3&ex=1114056000&pagewanted=all&position=

One Hundred Years of Uncertainty
By BRIAN GREENE

JUST about a hundred years ago, Albert Einstein began writing a paper that secured his place in the pantheon of humankind's greatest thinkers. With his discovery of special relativity, Einstein upended the familiar, thousands-year-old conception of space and time. To be sure, even a century later, not everyone has fully embraced Einstein's discovery. Nevertheless, say 'Einstein' and most everyone thinks 'relativity.'

What is less widely appreciated, however, is that physicists call 1905 Einstein's 'miracle year' not because of the discovery of relativity alone, but because in that year Einstein achieved the unimaginable, writing four papers that each resulted in deep and formative changes to our understanding of the universe. One of these papers - not on relativity - garnered him the 1921 Nobel Prize in physics. It also began a transformation in physics that Einstein found so disquieting that he spent the last 30 years of his life in a determined effort to repudiate it.

Two of the four 1905 papers were indeed on relativity. The first, completed in June, laid out the foundations of his new view of space and time, showing that distances and durations are not absolute, as everyone since Newton had thought, but instead are affected by one's motion. Clocks moving relative to one another tick off time at different rates; yardsticks moving relative to one another measure different lengths. You don't perceive this because the speeds of everyday life are too slow for the effects to be noticeable. If you could move near the speed of light, the effects would be obvious.

The second relativity paper, completed in September, is a three-page addendum to the first, which derived his most famous result, E = mc2, an equation as short as it is powerful. It told the world that matter can be converted into energy - and a lot of it - since the speed of light squared (c2) is a huge number. We've witnessed this equation's consequences in the devastating might of nuclear weapons and the tantalizing promise of nuclear energy.

The third paper, completed in May, conclusively established the existence of atoms - an idea discussed in various forms for millenniums - by showing that the numerous microscopic collisions they'd generate would account for the observed, though previously unexplained, jittery motion of impurities suspended in liquids.

With these three papers, our view of space, time and matter was permanently changed.

Yet, it is the remaining 1905 paper, written in March, whose legacy is arguably the most profound. In this work, Einstein went against the grain of conventional wisdom and argued that light, at its most elementary level, is not a wave, as everyone had thought, but actually a stream of tiny packets or bundles of energy that have since come to be known as photons.

This might sound like a largely technical advance, updating one description of light to another. But through subsequent research that amplified and extended Einstein's argument (see Figures 1 through 3), scientists revealed a mathematically precise and thoroughly startling picture of reality called quantum mechanics....

[Bysshe]

I have seen a lot written about the "unreasonable" effectiveness of math (and in fact my undergrad thesis was in this area of philosophy), but I really still can't see what is so shocking or unreasonable about the effectiveness of math. Math is really good at explaining things in math and science because we designed it that way. Formalisms turn out to be actual results because if we have to fudge an equation, there's usually an actual reason for the thing we introduce.

[Les Brunswick]

"Math is really good at explaining things in math and science because we designed it that way." That makes it sound similar to "a shoehorn is helpful for putting on shoes because we designed it to do that"

The problem is, why is the universe such that mathematics can be so extrordinarily effective at describing it? Did God design the universe so it could be so understood, and the human mind so that it could do mathematic? Why should the universe operate according to principles that can be described with symbols on a page? It is perfectly easy to imagine a universe in which this is not true. Or else it could be mathematically describable, but the mathematics beyond the comprehension of the human brain.

I was a physics major for a couple of years, and everyone considered it an unfathomable mystery as to why the universe is so mathematical.

[matt]

I'm always amazed that Planck's was a calculation in many body physics, something we normally put off until the second semester in a quantum mechanics course. So it's not quite correct to say that the first equation of quantum mechanics was around for years before it was recognized as such---it was actually Eq. (53).

Brad, if you read these comments, our own (UCB) Robert Littlejohn is currently teaching an upper-division seminar on the history of Planck's derivation. I don't think there's a course webpage, but he would probably email his notes if asked.

[anne]

http://www.nytimes.com/2005/04/08/opinion/08greene.html?ei=5070&en=0d30b3c8c6750ba3&ex=1114056000&pagewanted=all&position=

One Hundred Years of Uncertainty
By BRIAN GREENE

Before the discovery of quantum mechanics, the framework of physics was this: If you tell me how things are now, I can then use the laws of physics to calculate, and hence predict, how things will be later. You tell me the velocity of a baseball as it leaves Derek Jeter's bat, and I can use the laws of physics to calculate where it will land a handful of seconds later. You tell me the height of a building from which a flowerpot has fallen, and I can use the laws of physics to calculate the speed of impact when it hits the ground. You tell me the positions of the Earth and the Moon, and I can use the laws of physics to calculate the date of the first solar eclipse in the 25th century. What's important is that in these and all other examples, the accuracy of my predictions depends solely on the accuracy of the information you give me. Even laws that differ substantially in detail - from the classical laws of Newton to the relativistic laws of Einstein - fit squarely within this framework.

Quantum mechanics does not merely challenge the previous laws of physics. Quantum mechanics challenges this centuries-old framework of physics itself. According to quantum mechanics, physics cannot make definite predictions. Instead, even if you give me the most precise description possible of how things are now, we learn from quantum mechanics that the most physics can do is predict the probability that things will turn out one way, or another, or another way still.

The reason we have for so long been unaware that the universe evolves probabilistically is that for the relatively large, everyday objects we typically encounter - baseballs, flowerpots, the Moon - quantum mechanics shows that the probabilities become highly skewed, hugely favoring one outcome and effectively suppressing all others. A typical quantum calculation reveals that if you tell me the velocity of something as large as a baseball, there is more than a 99.99999999999999 (or so) percent likelihood that it will land at the location I can figure out using the laws of Newton or, for even better accuracy, the laws of Einstein. With such a skewed probability, the quantum reasoning goes, we have long overlooked the tiny chance that the baseball can (and, on extraordinarily rare occasions, will) land somewhere completely different.

When it comes to small objects like molecules, atoms and subatomic particles, though, the quantum probabilities are typically not skewed. For the motion of an electron zipping around the nucleus of an atom, for example, a quantum calculation lays out odds that are all roughly comparable that the electron will be in a variety of different locations - a 13 percent chance, say, that the electron will be here, a 19 percent chance that it will be there, an 11 percent chance that it will be in a third place, and so on. Crucially, these predictions can be tested. Take an enormous sample of identically prepared atoms, measure the electron's position in each, and tally up the number of times you find the electron at one location or another. According to the pre-quantum framework, identical starting conditions should yield identical outcomes; we should find the electron to be at the same place in each measurement. But if quantum mechanics is right, in 13 percent of our measurements we should find the electron here, in 19 percent we should find it there, in 11 percent we should find it in that third place. And, to fantastic precision, we do....

[anne]

http://www.nytimes.com/2005/04/08/opinion/08greene.html?ei=5070&en=0d30b3c8c6750ba3&ex=1114056000&pagewanted=all&position=

Einstein, radical thinker that he was, still believed in the sanctity of a universe that evolved in a fully definite, fully predictable manner. If, as quantum mechanics asserted, the best you can ever do is predict probabilities, Einstein countered that he'd "rather be a cobbler, or even an employee in a gaming house, than a physicist."

This emphasis, however, partly obscures a larger point. It wasn't the mere reliance on probabilistic predictions that so troubled Einstein. Unlike many of his colleagues, Einstein believed that a fundamental physical theory was much more than the sum total of its predictions - it was a mathematical reflection of an underlying reality. And the reality entailed by quantum mechanics was a reality Einstein couldn't accept.

An example: Imagine you shoot an electron from here and a few seconds later it's detected by your equipment over there. What path did the electron follow during the passage from you to the detector? The answer according to quantum mechanics? There is no answer. The very idea that an electron, or a photon, or any other particle, travels along a single, definite trajectory from here to there is a quaint version of reality that quantum mechanics declares outmoded.

Instead, the proponents of quantum theory claimed, reality consists of a haze of all possibilities - all trajectories - mutually commingling and simultaneously unfolding. And why don't we see this? According to the quantum doctrine, when we make a measurement or perform an observation, we force the myriad possibilities to ante up, snap out of the haze and settle on a single outcome. But between observations - when we are not looking - reality consists entirely of jostling possibilities.

Quantum reality, in other words, remains ambiguous until measured. The reality of common perception is thus merely a definitive-looking veneer obscuring the internal workings of a highly uncertain cosmos. Which is where Einstein drew a line in the sand. A universe of this sort offended him; he could not accept, as he put it, that "the Old One" would so profoundly incorporate a hidden element of happenstance in the nature of reality. Einstein quipped to his quantum colleagues, "Do you really think the Moon is not there when you're not looking?" and set himself the Herculean task of reworking the laws of physics to resurrect conventional reality....

Brian Greene, a professor of physics and mathematics at Columbia, is the author of “The Elegant Universe,’’ and, most recently, “The Fabric of the Cosmos.”

[larry birnbaum]

I haven't thought deeply about this, so no doubt this is horribly naive. But I don't see this as unreasonable efficacy -- if the world is lawful (i.e., same conditions lead to same results) and objects, systems, attributes have measurable quantity, not just kind, this is what happens. It is of course a mystery why these are the case but I don't see that as being about math per se.

And it really isn't a mystery that people don't know all the implications of what they are saying when they write down an equation. They don't know all of the implications of what they are saying regardless of what they are talking about or what language they are using. For example, when they write laws. That's part of the reason we need courts.

[dilbert dogbert]

I have been reading Eric Temple Bell's "Mathematics, Queen and Servant of Science". It is accessable to people like me, engineer educated in the 50' and 60's so a lot of the newer non engineers/science majors should find it accesable too. In his introduction he quotes H.R. Hertz's remark that: :It seems as if the mathematical implements we use are wiser than we, and perform their evolutions independently of our will." The other quote I found interesting was from the dispute between Fourier and Jacobi, where Jacobi responding to Fourier's distain of pure mathematics, said: "a scientist of Fouriers caliber should know that the true end of mathematics is the greater glory of the human mind."

[Aaron Bergman]

I have seen a lot written about the "unreasonable" effectiveness of math (and in fact my undergrad thesis was in this area of philosophy), but I really still can't see what is so shocking or unreasonable about the effectiveness of math. Math is really good at explaining things in math and science because we designed it that way. Formalisms turn out to be actual results because if we have to fudge an equation, there's usually an actual reason for the thing we introduce.

This would have more chance of being true if the math followed or was concurrent with the physics. In fact, it's usually the other way around. General relativity was based on ideas of curvature that went back to Riemann and before. The most impressive example for me is gauge theories (including electromagnetism) which goes back to Cartan, long predating their physics discovery (other than electromagnetism, of course). Actually, I'd be curious to know if Cartan knew that Maxwell's equations can be formulated in terms of connections and bundles. That the forces of nature should be based on abstract Lie groups seems utterly bizarre. And yet, it's true.

[Nicholas Weininger]

Don't forget EPR pairs-- Einstein thinks 'em up as a *counterexample*, saying "quantum mechanics can't be right because if it were, these crazy things would exist, and they obviously don't", and then we find 'em.

[Ben V-L]

I'm a theoretical physicist, and I have a little trouble finding the effectiveness of mathematics unreasonable. More surprising to me is the reproducibility of nature: if we take the effort to control conditions well enough, we can evidently be assured of a certain result (this even applies to quantum mechanics: it predicts certain probabilities, and these can be measured). This is surprising: the Greeks never really conceived of it being possible, and it took the likes of Bacon, Galileo, and ultimately Newton to create the scientific method and demonstrate that it had some genuine usefulness.

Once the reproducibility is established, the math just becomes a concise language for talking about it.

[anne]

Nicholas Weininger: EPR pairs.

http://www.nytimes.com/2005/04/08/opinion/08greene.html?ei=5070&en=0d30b3c8c6750ba3&ex=1114056000&pagewanted=all&position=

In 1935, through a disarmingly simple mathematical analysis, Einstein (with two colleagues) established a beachhead on the first front. He proved that quantum mechanics is either an incomplete theory or, if it is complete, the universe is - in Einstein's words - "spooky." Why "spooky?" Because the theory would allow certain widely separated particles to correlate their behaviors perfectly (somewhat as if a pair of widely separated dice would always come up the same number when tossed at distant casinos). Since such "spooky" behavior would border on nuttiness, Einstein thought he'd made clear that quantum theory couldn't yet be considered a complete description of reality....

Decades of painstaking experimentation have confirmed quantum theory's predictions beyond the slightest doubt. Moreover, in a shocking scientific twist, some of the more recent of these experiments have shown that Einstein's "spooky" processes do in fact take place (particles many miles apart have been shown capable of correlating their behavior). It's a stunning finding, and one that reaffirms Einstein's uncanny ability to unearth features of nature so mind-boggling that even he couldn't accept what he'd found....

[anne]

Ben V-L: Terrific :)

"I'm a theoretical physicist, and I have a little trouble finding the effectiveness of mathematics unreasonable. More surprising to me is the reproducibility of nature: if we take the effort to control conditions well enough, we can evidently be assured of a certain result (this even applies to quantum mechanics: it predicts certain probabilities, and these can be measured)....

"Once the reproducibility is established, the math just becomes a concise language for talking about it."

[Jennifer]

Ben V-L could you please continue your thought? Thanks for iterating it, Anne. How interesting.

[Jennifer]

The Brian Greene article is also wonderful. Thanks for posting it Anne.

[Ben V-L]

I would be happy to continue my comment, though I'm not sure in which direction. Let me bring it back to Brad's comments:

'It is this "what if we took this equation seriously?" factor that is, to my mind at least, the spookiest thing about the unreasonable effectiveness of mathematics in physics. Take the h in Max Planck's equation seriously, and you have the quantum principle--something that was not in Planck's brain when he wrote the equation down.'

followed by the Maxwell Eq./Special Relativity connection, etc. These are, of course, remarkable discoveries. But I don't find the route to discovery as remarkable as Brad does. A good analog to theoretical physics is a crossword puzzle. There are a lot of clues, some we can understand with great certainty and some we have a guess for their meaning, and some we have no idea. But they are also interdependent, so when we find a consistency between 2 or 3 of our shaky answers, they mutually gain credibility (the crossword overlap). And sometimes a letter firmly planted into a difficult clue provides the catalyst for finally understanding the clue - that's how I read Brad's examples.

Planck got his 'h' by fair means: he was lucky that his presumed quantization contained more reality than he expected, but he followed through its consequences and came up with a blackbody spectrum that matched empirical data. That made 'h' significant. He did have some awareness that this was significant but he could only plant a letter firmly (literally, sometimes analogies go too far!). It appears to me that Brad is surprised that this firmly planted letter happened to find itself in one of the longer and more difficult words of the puzzle. I don't see why this is suprising. Math is more or less irrelevant to the point: it's only a concise language for logical reasoning.

On the other hand, we proceed these days with a great hubris: I assume that if I send my students upstairs to measure Planck's constant, that they should find the electrons and photons to behave the same way everyone else has, and that they should come back to me with the same value everyone has found (absent the accuracy, since they would be doing the big money measurements). I am very comfortable with this assumption, because I come after the scientific revolution so I can point to a large number of cases where the reproducibility of nature has been demonstrated (the first definitive example: Newton finding a universal graviation law and a law of mechanics that together explained everything Brahe and Kepler had measured/analyzed in the solar system). Before these examples existed, it was not a natural thing to assume: the Greeks saw nature as a wild, unpredictable place, because it is until you control enough variables.

This kind of empirical reality - we can go ask nature the value of h whenever we like and it will obligingly tell us the same thing - makes science as we know it possible, and makes possible the discoveries and the connections between the discoveries and the math we explain them with.

That, in my amateur opinion, is why the Greeks didn't have an industrial revolution. They were way too far from having the proper appreciation of empirical reality.

[P.M.Lawrence]

One intriguing thing is that the universe at least appears to be UNmathematical, in the sense that it lazily discards a lot of mathematical things and behaves more sensibly than it needs to. That's the opposite of the nuttiness of quantum physics and the like.

To give just one example, when I read Mathematics at Cambridge, I remember one lecturer describing how to use certain techniques to solve physical problems. With tongue in cheek, he said "As all physicists know, all operators are Hermitian." At that everybody burts out laughing (well, you had to be there).

The point is, mathematics finds pathological objects and behaviours that don't SEEM to turn up in real life. Is it that thay don't, that it takes time for them to show up (like Hardy's bost that his three favourite discoveries weren't practical - yet all have now found uses), or is it that there is always a front with mathematics pushing ahead of people discovering physical reality by using the available mathematics?

I suppose I'm really saying that the game's not over so all we have so far is a partial result. As, when and if Lovecraftian stuff bursts out of the cosmos, its essentially ILlogical nature nay be revealed. How would we know otherwise, before the event? And maybe anthropocentric facts prevent any such reality ever emerging - a variation on cosmic self censorship manufacturing a mere appearance of logical and mathematical structure.

But wait - self censorship and anthropocentric arguments are logical... My brain hurts.

[anne]

Ben V-L

Certainly you developed your thought in an especially inciteful direction :) Let me try to think along.

Though I have heard it said that it is surprising and even unreasonable how effective mathematics is in the sciences, why should I be surprised? If you begin with nature and look to nature systematically or empirically, if you look to nature with rigorous method whether you are Newton or Darwin, there will be found relationships and regularities. What after all is the language of relationships and regularities? Mathematics is such a language.

The miracle of modernism is then the grown sense that nature can and should be looked at not singularly or with each glance being discrete, but nature can be looked to systematically. We can organize reproducible conditions, look repeatedly at a physical landscape and generalize about what we find.

Scientific method is the wonder that allows for translating observations or anticipated observations to mathematics. Wonderful, indeed.

[anne]

These ideas are not properly clear as yet, but I will play. As physical scientists have found reproducible relationships and regularities in nature and set these regularities in mathematical terms philosophers and social scientists have sought to do the same. Kant sought a regularity in ethics or a mathematics of ethics. "Critique of Pure Reason" is based on Newtonian Laws and the theoretical regularity is most engaging and persuasive. But when Kant sought to describe biological nature with Newtonian concepts there was of course no success.

So, Kant looked away to Aristotle and allowed for a discreteness in the biological world but a discreteness that was directed to an end. We have a teleology to biology. Of course this is not so, as Darwin showed conclusively. Since Kant we really are looking for a philosophy of biology, biologically based, and being at the same time afraid to look. If Einstein resisted accepting a probabilistic universe, we resist more so population thinking as Darwin understood was necessary.

[anne]

Ben V-L thank you, and thanks for every comment on this remarkable thread :)

[Alejandro Rivero]

My candidate as the real father of the revolution is Arnold Sommerfeld. He got the fine structure constant before the advent of new quantum mechanics, by combining the quanta postulate, Bohr theory, and relativity.

If you look at
http://www.physcomments.org/wiki/index.php?title=Genealogy::nobel
you will notice I have not been able to trace a direct heritage of Max Plank. At 1915 Born was appointed to assist Planck, but this relationship was truncated because of the war.

[anne]

The essentialist thinking of Plato would make reproducibility of nature irrelevant. All that is necessary is to recognize a class of objects. Tree, or oak or evergreen tree is sufficient classification. There is an essential treeness that can be readily recognized but describing a population of trees is not necessary.

Ben V-L

"More surprising to me is the reproducibility of nature: if we take the effort to control conditions well enough, we can evidently be assured of a certain result (this even applies to quantum mechanics: it predicts certain probabilities, and these can be measured). This is surprising: the Greeks never really conceived of it being possible, and it took the likes of Bacon, Galileo, and ultimately Newton to create the scientific method and demonstrate that it had some genuine usefulness."

[anne]

http://www.edge.org/3rd_culture/mayr/mayr_index.html

ERNST MAYR: WHAT EVOLUTION IS

MAYR: One of my themes is that Darwin changed the foundations of Western thought. He challenged certain ideas that had been accepted by everyone, and we now agree that he was right and his contemporaries were wrong. Let me just illuminate some of them. One such idea goes back to Plato who claimed that there were a limited number of classes of objects and each class of objects had a fixed definition. Any variation between entities in the same class was only accidental and the reality was an underlying realm of absolutes.

EDGE: How does that pertain to Darwin?

MAYR: Well Darwin showed that such essentialist typology was absolutely wrong. Darwin, though he didn't realize it at the time, invented the concept of biopopulation, which is the idea that the living organisms in any assemblage are populations in which every individual is uniquely different, which is the exact opposite of such a typological concept as racism. Darwin applied this populational idea quite consistently in the discovery of new adaptations though not when explaining the origin of new species.

[Foye]

Gosh, I love having happened upon all this physics history talk: Mathematics mirroring reproducibility in physics . . . quantum mechanics . . . Darwin, even.

Ah, the mind wanders into wild speculation. Of course there's reproducibility. All laws of nature which didn't or don't provide systemic sustainability, (including any weird stuff which might occasionally still pop up), become prompt losers in the contest for survival of the fittest natural laws. Shall Darwin's ideas be applied to non-biological existence? Look for an anomaly. (Of course, if it's non-reproducible . . . )

[Bruce Garrett]

"It is this "what if we took this equation seriously?" factor that is, to my mind at least, the spookiest thing about the unreasonable effectiveness of mathematics in physics."

I recall many years ago, watching some TV science show on the beginnings of radio. At the beginning of it there was this story to the effect that someone working with equations on magnetism and electricity (was it Maxwell?) found the equation they were working with unbalanced and didn't like it, and for what was suggested as being purely artistic reasons, added something to them called a displacement current, which made the equation look more balanced and beautiful. And then someone else came along later looked at this displacement current thing and began building experimental radio transmitters.

I probably have the details of this one all mucked up, but it's always been the thing I've thought of when pondering physics and mathematics and beauty.

[larry birnbaum]

Look, there are lots of things that we can say "what if we actually took this seriously?" about that aren't mathematical and for which the exercise leads to profound and interesting consequences. For example, "We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness." It's interesting, but it's got nothing to do with math.

[john c. halasz]

Just to add a view from a bleacher-bum, mathematics is the science of formal operations. It elaborates "systematically" entirely formal rules of implicature. When new "things" are discovered in the development of mathematics, such as, say, the concept of "zero",- (and, n.b., Greco-roman mathematics lacked that concept, so it would be hard to imagine them elaborating much by way of a mathematicized physics), or imaginary numbers, or complex numbers,etc., it is because they are "seen" to make some sort of mathematical "sense", that is, they open upon new domains of formal implicature that can be recombined with already understood domains and further elaborated upon, whereby means are derived by which they can be "proven". (Isn't this something of the sense of Goedel's Incompleteness Theorem, that there are always mathematical "entities" "out there" that can't be proven?) Now, there is a virtual infinity of mathematical patterns that can be elaborated, and it is by no means clear in what sense they should be taken. (My favorite story in that regard is of the Medieval Chinese mathematician who authored a treatise on algebra, elaborating numerous methods, all of which had the same answer. Presumably the "Western" approach would have been to reduce to a unity of method that produced variable answers.) But the point about the application of mathematics in science is that it is an *application*; of the innumerable possible patterns in mathematics, these particular ones are found to apply. It is not that nature-in-itself is somehow intrinsically mathematical; it is that we find the patterns that fit the data. The justification of such a procedure, its empirical content by which we can frame and serially systematize explanations of phenomena, consist in the physical constants discovered to obtain, not in the mathematical formalism itself. And that justification of projected explanations involves, as well, a substantive delimitation and interpretation of the domain of phenomena or objects, together with a process of abstraction, by which parameters suceptible to quantititive methods can be isolated. (The 17th century distinction between primary and secondary qualities, which actually makes no sense, as if mass or volume were not sensory properties in the first instance, or as if color were not a physical property, is nonetheless a tell-tale marker of such a process of interpretation and abstraction.) Now all rational studies that get established as sciences involve a threshhold of formalization, as one of their hallmarks, but, pace Kant, that does not make mathematics the sine qua non of science, since such formalizations might not be quantitative in nature, as with, e.g., cladistics in biology. Such a notion commits the fallacy that mathematical-theoretical physics, as the paradigmatic science, is the "essence" of all science, since all other processes are dependent of the existence and dynamics of matter and energy, which misses, of course, that that is a necessary, but not sufficient condition for the delimitation and understanding of other domains of phenomena. (And, of course, some disciplines, which merely pretend to be science, such as economics, make extensive use of mathematical methods. But there the mathematical dimension is already partly constitutive of the real subject-matter, as it concerns the ennumerative and calculative activities of human beings, as well as, the physical constraints encountered in material processes of labor. Going back to the recent debate here over Marx' deployment of LTV, one of its basic points was to explain, in terms of conditions of possibility, how incomparable objects and activities can be construed as quantitatively identical. And Alfred Sohn-Rethel, among others, has argued that the revival and extension of trade, with the resulting penetration of quantitative methods and practical problems and techniques, was the empirical basis for the "transcendental" constitution of science.) But the main point here is that the application of mathematical methods in different fields of science involves different frameworks and knowledge-constitutive rules defining the respective domains of explanation and the parameters and aims involved, such that there would be subtle differences in understandings, not just between physicists and mathematicians, but between physicists and chemists, chemists and geologists, etc. That stated, the fundamental basis of science is the observation of regularities in nature or the world. And the ultimate basis of the so-called "laws of nature", which the various sciences attempt to serially systematize, is the statistical distribution of those regularities. I wouldn't know how to answer the question as to whether those statistical distributions actually exist "in" nature or not, or whether that is even an intelligible question. What can be said is that such a statistical view of the "laws of nature" was not possible to conceive before the latter half of the 19th century, when mathematical statistics was sufficiently elaborated to begin to formulate such a view.

I found the above attribution of a "wild" mentality to the ancient Greeks, by which they could not form a "truly" empirical science rather odd, not to mention at odds with the standard, received account that the rational thought-form of Greek metaphysics formed a significant precursor in the forecourt of the eventual emergence of modern science. To begin with, it should be pointed out that Plato was far more "Aristotelian" than his traditional "otherworldly" reputation, transmitted through later Neo-Platonism, would allow for, just as Aristotle's sometimes perplexing criticisms of Plato, starting with the deployment of the existence/essence distinction against the doctrine of Ideas, really amount to doing Plato one better. But not just the "discovery" of a quasi-systematic rational implicature in the categorical structure of thought, as the hallmark of post-mythical worldly reality, but the very "ousiological" notion that the world is somehow, rather mysteriously, transparently intelligible were essential to the formation of modern science and "Enlightenment" and at least partly continuous with it. And Aristotle, in particular, is somewhat anachronistically credited with having established empirical observation as the means for the extension of knowledge. (My favorite story about Aristotle the empiricist concerns eels. Aristotle thought that most animal species reproduced from seed, like following from like, in accordance with the categorical structure of the cosmos. But some species he thought derived from spontaneous generation, one of which was eels. The actual empirical facts are these: Mediteranean eels do not develop sex organs until shortly before spawning, and after they spawn, they die...and they spawn off the coast of Bermuda.) But there were, of course, discontinuities, not necessarily of a rational nature, in the fall of the classical world into the "Dark Ages". For one thing, Christian doctrine conceived of the world as "fallen", which is to say, as sinful and therefore "dead". This gave to Medieval thinking a "materialist" cast that was to re-emerge in the formation of modern science on the basis of an understanding of nature as inanimate, for all that the new science had to fight off charges of "Epicurianism" and heresy. By contrast, for example, Aristotle conceived of motion as a property of moving bodies, in part, because his basic paradigm of naturally existent beings was the animal organism, and correspondingly he saw the cosmic process as a living thing. One might say that such a view is insufficiently differentiated and involves categorical confusions, but, on the one hand, it is by no means obvious that the Cartesian view animal organisms as purely mechanical entities, nor its subjectivist splitting-off of intelligibility are exactly an advance, while, on the other hand, if one were to imagine starting from scratch, as it were, it is by no means obvious that the Aristotelian view is incorrect. The other important transformation was that modern science emerged partly from the influence of nominalism, which was itself a product of the bifurcation and decay of the understanding of Aristotelian thought in Medieval Scholasticism. (And, of course, the empiricist philosophy that is dominant in Anglo-Saxon cultures is a lineal descendent of Scholastic nominalism.) But if the new spirit of experimentation was, indeed, the decisive breakthough, it is by no means clear that it obviated the perplexities of thought, that "essences" could be simply dissolved into the plurality of phenomena without regard to the conceptual structure of their identification and demarcation.

As a last comment, that the formalism of Kantian ethics was "Newtonian" I would regard as a criticism, a misrecognition amidst the genuine recognitions of that work.

[anne]

John C. Halasz

Clever argument, which I will think about for a while :)

[anne]

http://www.nytimes.com/2005/02/14/arts/14conn.html?ei=5070&en=493b86d4d721055e&ex=1114142400&pagewanted=all&position=

Truth, Incompleteness and the Gödelian Way
By EDWARD ROTHSTEIN

Relativity. Incompleteness. Uncertainty.

Is there a more powerful modern Trinity? These reigning deities proclaim humanity's inability to thoroughly explain the world. They have been the touchstones of modernity, their presence an unwelcome burden at first, and later, in the name of postmodernism, welcome company.

Their rule has also been affirmed by their once-sworn enemy: science. Three major discoveries in the 20th century even took on their names. Albert Einstein's famous Theory (Relativity), Kurt Gödel's famous Theorem (Incompleteness) and Werner Heisenberg's famous Principle (Uncertainty) declared that, henceforth, even science would be postmodern.

Or so it has seemed. But as Rebecca Goldstein points out in her elegant new book, "Incompleteness: The Proof and Paradox of Kurt Gödel" (Atlas Books; Norton), of these three figures, only Heisenberg might have agreed with this characterization.

His uncertainty principle specified the inability to be too exact about small particles. "The idea of an objective real world whose smallest parts exist objectively," he wrote, "is impossible." Oddly, his allegiance to an absolute state, Nazi Germany, remained unquestioned even as his belief in absolute knowledge was quashed.

Einstein and Gödel had precisely the opposite perspective. Both fled the Nazis, both ended up in Princeton, N.J., at the Institute for Advanced Study, and both objected to notions of relativism and incompleteness outside their work. They fled the politically absolute, but believed in its scientific possibility.

And therein lies Ms. Goldstein's tale. From the late 1930's until Einstein's death in 1955, Einstein and Gödel, the physicist and the mathematician, would take long walks, finding companionship in each other's ideas. Late in his life, in fact, Einstein said he would go to his office just to have the "privilege" of walking with Gödel. What was their common ground? In Ms. Goldstein's interpretation, they both felt marginalized, "disaffected and dismissed in profoundly similar ways." Both thought that their work was being invoked to support unacceptable positions.

Einstein's convictions are fairly well known. He objected to quantum physics and its probabilistic clouds. God, he famously asserted, does not play dice. Also, he believed, not everything depends on the perspective of the observer. Relativity doesn't imply relativism.

The conservative beliefs of an aging revolutionary? Perhaps, but Einstein really was a kind of Platonist: He paid tribute to science's liberating ability to understand what he called the "extra-personal world."

And Gödel? Most lay readers probably know of him from Douglas R. Hofstadter's playful best-seller "Gödel, Escher, Bach," a book that is more about the powers of self-referentiality than about the limits of knowledge. But the latter is the more standard association. "If you have heard of him," Ms. Goldstein writes, perhaps too cautiously, "then there is a good chance that, through no fault of your own, you associate him with the sorts of ideas - subversively hostile to the enterprises of rationality, objectivity, truth - that he not only vehemently rejected but thought he had conclusively, mathematically, discredited." ...

[anne]

http://www.nytimes.com/2005/02/14/arts/14conn.html?position=&ei=5070&en=493b86d4d721055e&ex=1114142400&pagewanted=print&position=

Before Gödel's incompleteness theorem was published in 1931, it was believed that not only was everything proven by mathematics true, but also that within its conceptual universe everything true could be proven. Mathematics is thus complete: nothing true is beyond its reach. Gödel shattered that dream. He showed that there were true statements in certain mathematical systems that could not be proven. And he did this with astonishing sleight of hand, producing a mathematical assertion that was both true and unprovable.

It is difficult to overstate the impact of his theorem and the possibilities that opened up from Gödel's extraordinary methods, in which he discovered a way for mathematics to talk about itself. (Ms. Goldstein compares it to a painting that could also explain the principles of aesthetics.)

The theorem has generally been understood negatively because it asserts that there are limits to mathematics' powers. It shows that certain formal systems cannot accomplish what their creators hoped.

But what if the theorem is interpreted to reveal something positive: not proving a limitation but disclosing a possibility? Instead of "You can't prove everything," it would say: "This is what can be done: you can discover other kinds of truths. They may be beyond your mathematical formalisms, but they are nevertheless indubitable."

In this, Gödel was elevating the nature of the world, rather than celebrating powers of the mind. There were indeed timeless truths. The mind would discover them not by following the futile methodologies of formal systems, but by taking astonishing leaps, making unusual connections, revealing hidden meanings.

Like Einstein, Gödel was, Ms. Goldstein suggests, a Platonist....

[anne]

John C. Halasz

Kant is a wonder of humane ethical thinking and I am thoroughly respectful of the ethical approach extending from Kant to Rawls, but an essentialist thinking of Kant limits us in contingent understanding of the living world which surrounds and here we must go beyond what Kant imagined. There is much to be added.

[Jennifer]

Ben and Anne and John and PM Lawrence, et al...

Thank you all for this lovely stream of comments and articles. There is no more exciting blog, and I look forward each day to posts and comments.

[john c. halasz]

Anne:

My last comment was a bit cryptic, but I think the "Newtonian" derivation of the formalism of Kant's ethics is correct. Kant conceives of the phenomenal world entirely in terms of Newtonian causal determinism, such that "free will", which morality must presuppose, according to Kant, is "noumenal", unknowable as such. At the same time, he declares that a lawless will is a contradiction in terms and would not be free, since he implicitly conceives of "free will" in terms of causal efficacy; therefore he arrives at the purely formal nature of the moral law, as at once analogous to and disjunct from causal law. It should be pointed out that there is a deep systematic connection between the first and second "Critiques". Not only does Kant ground the normativity of knowledge epistemologically in the will, in a sort of "will to knowledge", which must have an ethical dimension, and not only does he turn to practical reason in an attempt to resolve something of the antinomian gaps left dangling in the "Critique of Pure Reason", but he is searching for the pure rational will, defined as the will which always wills itself, which he finds in the moral "good will" as the only thing that is good in-itself, and hence an end-in-itself. Hence supervenes the moment of the metaphysics of "presence", something unthinkable that grounds thinking. (Compare, e.g., the Aristotelian godhead, which, as thought thinking itself, is defined as pure actuality.) This leads to a severely driven conception of moral action, as well, as a marked intentionalist bias, in accordance with German/Lutheran "inwardness". That criticism stated, Kant's enduring legacy, in my book, has to due with his recognition of the de-ontic status of morality, and the unconditional character of its commitments. But I don't know whether the formalism/intellectualism of Kantian morality is "humane" or not; that would seem to me to depend on its application, which rather undermines its point.

I myself am something of a Wittgensteinian, a post-epistemological orientation, with an interest in the revival of the pre-modern Aristotelian distinction of practical reason from theoretical reason, which their Kantian conflation rather squelched. But I certainly don't claim to have all sorted out, though I do regard Kant as among the greatest of our precursors and opponents.

[anne]

John C. Halasz

Agreed :) Nicely done.

[anne]

http://www.edge.org/3rd_culture/mayr/mayr_index.html

ERNST MAYR: WHAT EVOLUTION IS

Another idea that Darwin refuted was that of teleology, which goes back to Aristotle. During Darwin's lifetime, the concept of teleology, or the use of ultimate purpose as a means of explaining natural phenomena, was prevalent. In his Critique of Pure Reason, Kant based his philosophy on Newton's laws. When he tried the same approach in a philosophy of living nature, he was totally unsuccessful. Newtonian laws didn't help him explain biological phenomena. So he invoked Aristotle's final cause in his Critique of Judgement. However, explaining evolution and biological phenomena with the idea of teleology was a total failure.

To make a long story short, Darwin showed very clearly that you don't need Aristotle's teleology because natural selection applied to bio-populations of unique phenomena can explain all the puzzling phenomena for which previously the mysterious process of teleology had been invoked.

The late philosopher, Willard Van Orman Quine, who was for many years probably America's most distinguished philosopher — you know him, he died last year — told me about a year before his death that as far as he was concerned, Darwin's greatest achievement was that he showed that Aristotle's idea of teleology, the so-called fourth cause, does not exist....

[anne]

John

"I myself am something of a Wittgensteinian, a post-epistemological orientation, with an interest in the revival of the pre-modern Aristotelian distinction of practical reason from theoretical reason, which their Kantian conflation rather squelched."

Of course, this you will have to clarify and justify for I wonder if even in the arts there is any need for the distinction :) Think of Cezanne's landscapes and the distinction may become artificial.

[anne]

So you think you want to separate practical and theoretical reason, do you?

http://www.nytimes.com/2005/04/15/books/15salamon.html?pagewanted=all&position=

In "The Art of Maurice Sendak," published in 1980, Selma G. Lanes quotes the artist saying: "There are basically two approaches to illustration. First, there is the direct, no-nonsense approach that puts the facts of the case into simple, down-to-earth images: Miss Muffet, her tuffet, curds, whey, spider and all. Then there is, for want of a better term, illumination. As with a poem set to song, in which every shade and nuance is given greater meaning by the music, so pictures can interpret texts."

[anne]

And, no one, no one at all, had better say a single word against Maurice Sendak :) Careful.

[john c. halasz]

Anne:

Don't haved much time now, so very briefly, the epistemological turn in modern philosophy conflated ethics with cognition, whereas they are distinct normative dimensions, "truth" being distinct from "rightness, justice, goodness". The old distinction between theoretical and practical reason amounts to theoretical reason solving problems within a more-or-less defined problematic with the aim toward the production/ascertainment of cognitive truth, whereas practical reason is not oriented toward solving "problems" but resolving conflicts, whether within or between human agents. To say that 'truth" and "justice" are distinct dimensions amounts to saying that in the real world they are at once cross-implicated and opposed.

Just to add a bit I forgot, that Kantian moment of the metaphysics of "presence" qua "the pure rational will which always wills itself" itself has something of a Newtonian ring to it qua "the constant action of forces". Also, I might point out that there is an alternative in the excluded middle between metaphysical teleology and causal immanence, namely, the systems-theoretic notion of teleonomic functionning.

[anne]

John

Understood and agreed. That was so nice of you. There will be more as there is time :) Clever argument.

[Anarch]

"Once the reproducibility is established, the math just becomes a concise language for talking about it."

Let me try to rephrase the issue to see if I can't rekindle the spookiness.

Consider the following syntactic operation: A, A->B : B.

This is a purely syntactic operation. It has an intended semantic meaning, of course -- this is modus ponens, or implication -- but that's irrelevant. It's a way of turning expressions into other expressions.

Consider the letter E. This too has an intended semantic meaning -- "membership", usually denoted by 'epsilon' by everyone but set theorists -- but that's also irrelevant. It's just a letter.

Now, consider the following two propositions:

1) For every observable phenomenon in the world -- every physical "truth", if you don't mind me waxing a little poetic -- there are a string of symbols (involving only 'E' and a few other symbols like '->') that represent this phenomenon.

That's not particularly surprising, I suppose. "Representation" is a particularly weak notion without some kind of restriction on what it means to "represent". But...

2) For every pair of observable phenomena in the world of the form A and A->B, the string of symbols B is also a physical truth.

IOW: the truths of the physical world are closed under a *syntactic* operation on *abstract strings of symbols*.

That, to me, is utterly mind-blowing. If you give me enough strings, I can, by completely abstract syntactic operations, transform them into a truth that I might never have observed, might never observe and, indeed, might never be *able* to observe. How in the world can the, well, world operate under such a restriction? It makes no sense whatsoever; IMO the only reason it doesn't freak out more people is that we're so saturated in it that they cannot see how utterly bizarre and restrictive it is.

[anne]

Anarch

'If you give me enough strings, I can, by completely abstract syntactic operations, transform them into a truth that I might never have observed, might never observe and, indeed, might never be *able* to observe.'

Please, an example.

[viacondotti]

"That, in my amateur opinion, is why the Greeks didn't have an industrial revolution. They were way too far from having the proper appreciation of empirical reality."

There are lots of specific historical conditions that might explain why the Greeks did not have an industrial revolution. Here's just one: slave-owning societies lack an important incentive to mechanize.

Anyway, in light of the reality-based scientific achievements of Greeks like Archimedes, it's a bit of a canard to keep relying on Plato's airiness and Aristotle's legendary unwillingness to count his wife's teeth to explain why the Greeks didn't, in addition to formalizing geometry, making significant contributions to number theory, and measuring the circumference of the earth, also tame electricity and develop the steam engine. One can only expect so much from a state whose largest city was smaller than Berkeley, CA.

Finally, surely there are future states of technology and historical turns in society to come that someone, looking back to the United States of the twentieth and twenty first century, might wonder how it took so long and how the obvious was missed....

[Ben V-L]

I don't underestimate the Greeks, by any means. In fact, it's their impressive progress in mathematics that makes their lack of physics (or better put, their wrong physics) more striking, and more demanding of explanation.

Let's put it this way: that the first overwhelming demonstration of the scientific method was planetary motion is no coincidence. In most mechanical systems there is a lot transfer of energy from mechanical to thermal (that is, friction), which was a considerable barrier in extracting the governing laws. But for planetary motion this is negligible - it's a clean system. The data was carefully taken, and the Greeks had a perfect understanding of geometry. This was the low-lying fruit (an apple?) that needed to be plucked so that they could proceed with confidence trying to apply the scientific method to complicated issues like phase transformations, kinetic theory of gasses, etc.

So to my mind, their failure to get as far as Newton in explaining the solar system shows (i) that they didn't just fail to find engines and electricity, they fell vastly short, failed to even start looking, and (ii) that there was likely some obstacle preventing them from making a discovery they were otherwise well-poised to make (the laws governing planetary motion).

[Ben V-L]

I left out my last sentence: In my amateur opinion, this obstacle was their cultural inclination to distrust empirical reality.

[Ben V-L]

Anarch, you make your case very clearly and elegantly, which gives me perhaps a better perspective on where Brad was coming from. But I still disagree about the significance, because discovery in physics doesn't work this way.

Let me give an example: Take 'A' to be Newtonian mechanics - a vastly successful theory that made many explicit predictions, many logical implications B1, B2, ... that indeed turned out to be true. Until we got around to probing the atomic scale, and then suddenly some of the implied consequencies A -> B gave B's that just weren't consistent with reality.

Solution: we go back and modify A. Now Newtonian mechanics is A', which is the same as before with some caveats about when we expect it to be valid and when not. (And we have A'', quantum mechanics, that gives a host of new predictions and appropriately reduces to A' in the right limits).

This is a fundamental maneuver in physics: the mathematical consequences of a theory (better put: the logical consequences of following the reasoning behind the construction of the theory) often turn out to be useful. This is because nature seems to have an internal structure, so we can in some sense find "truths" that we touch on with our theories. But when the logical consequences of a theory start not matching reality (i.e., the math leads to wrong predictions), we modify the theory. Nature is not compelled to modify its behavior to accomodate our wrong, previous understanding.

[anne]

Ben V-L

Thank you so much for following up. The matter is not defending the Greeks, but understanding links that might well have been made between their mathematics and radily observed phenomena. Links that were decidely not made. There simply was a vastly different approach to seeing than Newton understood and Hume fully generalized. A clever argument indeed. I am thinking this through.

[john c. halasz]

Well, I suppose this is just an illustration of the difference between thinking historically and thinking scientifically. "Energy" is itself a Greek-derivative, "energeia", which gets the latinate translation of "actuality", paired with "dumanis", likewise, "potentiality". The "ousia" was the power of cohesion a thing has of itself, which Aristotle conceived in terms a staged process of the unfolding of dumamis into energeia. In contrast to the Eleatic moment of the Platonic "eidos", Aristotle wanted to take account of "metabasis", "change", of which "kinesis" was a species. And he was especially interested, by way of understanding the "phusis" of the "kosmos", in the problem of "metabasis in allo genos", change into another kind, from which, somewhat mistakenly, the notion of "inductive" logic was subsequently read off. And speaking of logic, which Aristotle invented from the whole cloth, and the prestige of which has haunted us ever since, it is a question as to whether he conceived of its fundamental principle, "the law of the excluded middle", whereby a thing can not be and not be, nor an attribute and its opposite predicated of a thing at one and the same time, as a matter of formal thought, or whether that was not an "ontological" insight into the delimitation of the notion of "ousia", "substance", as the fundamental constituent of being, and "logic" was the methodical consolidation of that insight into the further extension of understanding. (Needless to say, we can not read a subject/object split back into Aristotle, as such a "problem" simply did not bother him.) And, of course, Aristotle subscribed to an astronomical theory, an earlier version of what was to be dubbed the Ptolemaic theory. (Incidentally, it's is a mistake to regard Aristotle as a monotheist: the godhead, qua unmoved mover, moves all things as the ultimate "telos telion", final end, because all things are moved by desire for it. But the first things so moved are the gods, conceived as embodied in the heavenly bodies, who are moved by pure desire, but who impart their motion to the sublunary sphere.) It's hard to imagine exactly why the ancients conceived of the heavens as a divine ether, but it was perhaps because they could neither manage, nor bear to conceive of the great cyclical ordering patterns of the heavens as a void rather than a significant, substantial reality.

The upshot here is that there are no facts, conveyed directly by sheer experience, outside of the frameworks in which those facts of "experience" occur, which co-determine their provenance and relevance. One can not simply leap over the historical development of frameworks, with their drive toward "rational" coherence even in their eroneousness, to arrive at the "true" facts. The hermeneutical task of understanding frameworks in their co-determination of facts remains to any cross-comparison of facts, since it is in fact part of the process of (understanding) how frameworks and their facts change. And that hermeneutical task remains relevant to the understanding of modern science(s), insofar as it is a matter of explicating the relationship between our ordinary experience and (practical) orientation in the world and the highly advanced and formalized explanatory understandings of contemporary science, which operate over the horizon of any ordinary human experience of the world, and the "translation" of its results, since the intelligibility of scientific explanations draw on the human experience of the world for their sense as much as science explains that experience. And I think it's worth pointing out that one of the differentiations involved in the emergence and development of modern science is that science does not aim at the ultimate "truth" of the universe, but rather aims at understanding the structure of real processes, which is an ongoing project whose provisional and revisable truths mix with and conflict with other truths and norms, even as the scope of such understanding is extended. But since the topic of this thread was basically the accounting for the role of (mathematical) formalisms in empirical understandings, I would want to emphasize again that the application of such formalisms are *applications* and that, in addition to those formalisms and the empirical "data", the role of substantive concepts that interpret specific domains of reality/experience of the world is crucial in warranting the application of those formalisms to the relevance of that data. And the sources and derivation of scientific concept formation, by which theories can be constructed with testable implications, is at least as interesting and crucial a topic as the structure and efficacy of formalizations. If I may reveal my biases, I think that Kant was half-right, after all, about the role of "synthetic judgments a priori" in the "constitution" of scientific knowledge, except that they are not nearly as invariant, timeless, ahistorical, unempirical, nor identifiable with actual direct experience and its "necessity" as he would have made them out to be.

The Newtonian definition of time as the measure of motion through space, now outmoded, was not unprecedented. Compare the Aristotelian definition: "the number of movement through change in place." Even as the substantialism of Aristotelian "science" was being displaced, (more by feathers and cannon balls than celestial matters), Newton erected time and space into a sort of absolutized substance containing events. Substantialism was drawn upon even as it was displaced, and that severe abstraction was too an artefact of its time. It was Einstein who "freed" us from the reification of that severe abstraction, with its arbitrary and seemingly reductionist implications, by understanding time as itself a physical parameter together with physical space and events, a feat of rational integration that was at the same time a further de-substantialization. Perhaps the irony is that the drive for rational integration, which once gave rise to the outmoded metaphysical form of thought, remains with us in the midst of our increasingly differentiated, de-substantialized and plural understandings of the world.