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June 28, 2008

Michael Polanyi and Personal Knowledge

Posted by David Corfield

There was a discussion over at the Secret Blogging Seminar about the differences between mathematics and the natural sciences, which interested me greatly as someone who has frequently looked to the philosophy of science for ideas about how to treat mathematics. By and large Anglophone philosophy has chosen to treat these disciplines very differently, and has overlooked opportunities to elaborate their similarities, such as furthering George Polya’s Bayesian treatment of mathematics.

One way to lessen the difference between the disciplines is to bring to centre stage the personal involvement of scientist and mathematician in their respective theories. In that each is a member of a tradition of long standing, each has to struggle against some intransigent reality, and to convince their colleagues that their perspective on this reality is a good one, they can be seen to have much in common. For some, however, the distinction between empirical evidence and whatever support a mathematician receives trumps any such consideration.

Michael Polanyi in Personal Knowledge, written in 1958, while reflecting on this latter difference, seeks to understand it in the context of a general account of participation in a wide range of practices:

The acceptance of different kinds of articulate systems as mental dwelling places is arrived at by a process of gradual appreciation, and all these acceptances depend to some extent on the content of relevant experiences; but the bearing of natural sciences on facts of experience is much more specifiable than that of mathematics, religion or the various arts. It is justifiable, therefore, to speak of the verification of science by experience in a sense which would not apply to other articulate systems. The process by which other systems than science are tested and finally accepted may be called, by contrast, a process of validation.

Our personal participation is in general greater in a validation than in a verification. The emotional coefficient of assertion is intensified as we pass from the sciences to the neighbouring domains of thought. But both verification and validation are everywhere an acknowledgement of a commitment: they claim the presence of something real and external to the speaker. As distict from both of these, subjective experiences can only be said to be authentic, and authenticity does not involve a commitment in the sense in which both verification and validation do. (p. 202)

Some participants in the Secret Blogging Seminar debate, such as Terence Tao, were keen to provide a continuum between mathematics and physics. An important point to note in this respect is that physics is guided by other than empirical considerations. Earlier in the book, pp. 9-15, Polanyi discusses how Einstein came to relativity theory more by way of what we have heard him call above validation than by verification. He continues:

When the laws of physics thus appear as particular instances of geometric theorems, we may infer that the confidence placed in physical theory owes much to its possessing the same kind of excellence from which pure geometry and pure mathematics in general derive their interest, and for the sake of which they are cultivated.(p. 15)

Again we are returned to the passionate, personal engagement of the scientist with their field. I’ll leave you with his diagnosis of the mistake which encourages us to discount this engagement:

We cannot truly account for our acceptance of such theories without endorsing our acknowledgement of a beauty that exhilarates and a profundity that entrances us. Yet the prevailing conception of science, based on the disjunction of subjectivity from objectivity, seeks–and must seek at all costs–to eliminate from science such passionate, personal, human appraisals of theories, or at least to minimize their function to that of a negligible by-play. For modern man has set up as the ideal of knowledge the conception of natural science as a set of statements which is ‘objective’ in the sense that its substance is entirely determined by observation, even while its presentation may be shaped by convention. This conception, stemming from a craving rooted in the very depths of our culture, would be shattered if the intuition of rationality in nature had to be acknowledged as a justifiable and indeed essential part of scientific theory. That is why scientific theory is represented as a mere economical description of facts; or as embodying a conventional policy for drawing empirical inferences; or as a working hypothesis, suited to man’s practical convenience–interpretations that all deliberately overlook the rational core of science.

That is why, also, if the existence of this rational core yet reasserts itself, its offensiveness is covered up by a set of euphemisms, a kind of decent understatement like that used in Victorian times when legs were called limbs–a bowdlerization which we may observe, for example, in the attempts to replace ‘rationality’ by ‘simplicity’. It is legitimate, of course, to regard simplicity as a mark of rationality, and to pay tribute to any theory as a triumph of simplicity. But great theories are rarely simple in the ordinary sense of the term. Both quantum mechanics and relativity are very difficult to understand; it takes only a few minutes to memorize the facts accounted for by relativity, but years of study may not suffice to master the theory and see these facts in its context. Hermann Weyl lets the cat out of the bag by saying: ‘the required simplicity is not necessarily the obvious one but we must let nature train us to recognize the true inner I simplicity.’ In other words, simplicity in science can be made equivalent to rationality only if ‘simplicity’ is used in a special sense known solely by scientists. We understand the meaning of the term ‘simple’ only by recalling the meaning of the term ‘rational’ or ‘reasonable’ or ‘such that we ought to assent to it’, which the term ‘simple’ was supposed to replace. The term ‘simplicity’ functions then merely as a disguise for another meaning than its own. It is used for smuggling an essential quality into our appreciation of a scientific theory, which a mistaken conception of objectivity forbids us openly to acknowledge.

What has just been said of ‘simplicity’ applies equally to ‘symmetry’ and ‘economy’. They are contributing elements in the excellence of a theory, but can account for its merit only if the meanings of these terms are stretched far beyond their usual scope, so as to include the much deeper qualities which make the scientists rejoice in a vision like that of relativity. They must stand for those peculiar intellectual harmonies which reveal, more profoundly and permanently than any sense-experience, the presence of objective truth.

I shall call this practice a pseudo-substitution. It is used to play down man’s real and indispensable intellectual powers for the sake of maintaining an ‘objectivist’ framework which in fact cannot account for them. It works by defining scientific merit in terms of its relatively trivial features, and making these function then in the same way as the true terms which they are supposed to replace.

Other areas of science will illustrate even more effectively these indispensable intellectual powers, and their passionate participation in the act of knowing. It is to these powers and to this participation that I am referring in the title of this book as ‘Personal Knowledge’. We shall find Personal Knowledge manifested in the appreciation of probability and of order in the exact sciences, and see it at work even more extensively in the way the descriptive sciences rely on skills and connoisseurship. At all these points the act of knowing includes an appraisal; and this personal coefficient, which shapes all factual knowledge, bridges in doing so the disjunction between subjectivity and objectivity. It implies the claim that man can transcend his own subjectivity by striving passionately to fulfil his personal obligations to universal standards.

Posted at June 28, 2008 9:24 AM UTC

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19 Comments & 1 Trackback

Dimensions of model; 2 forms of Knowing; Re: Michael Polanyi and Personal Knowledge

Two quick comments first, as deep thought ensues.

(1) Terry Tao, in suggesting a second axis in a chart of Math versus Science, namely concrete/abstract, replaces a spectrum analogy with a manifold – the Euclidean R^2. But why 2 axes? Why not R^n for some larger n?

(2) “Anglophone philosophy has chosen to treat these disciplines very differently…”

In part because the English language does not so clearly distinguish two forms of knowing: “knowing-that” and “knowing-how.”

French. Greek, Latin and Russian have constructions similar to “savoir faire” and “savoir comment faire”. German doesn’t, but the German translations for “knowing how” suggests why English goes astray..

In German, “wissen wie”, the most literal translation of “knowing how”, works pretty much the way the English construction (always) works. That is, it means savoir comment faire. Thus French sentences containing “savoir [faire]” shouldn’t be translated into German using “wissen wie”. Rather, one has to use the German “können”, which otherwise comes close to the English “can” (not to be confused with “kennen”).

I’ve said on earlier threads that there are 5 magesteria, each with a different apparatus for “truth” and for “proof.” Mathematics is (at least since Euclid) centered on Axiomatic Truth, and Science on Empirical Truth. The experimentalist is “knowing-how” in a way that Philsophy of Mathematics mostly denies to Math. But in the age of new foundations (including n-Catgeory Theory) and of Experimental Mathematics, the distinction does need to be re-appraised.

Posted by: Jonathan Vos Post on June 29, 2008 6:02 PM | Permalink | Reply to this

Re: Michael Polanyi and Personal Knowledge

The important point is, I think, something like “emergent phenomena”.

Inside the totality of chemistry, there happens to be a subfield, organic chemistry, in which structures appear that are vastly more complex than the rest of chemical phenomena. This way biology appears first inside chemistry.

Then inside biology there happens to be a subfield, human biology, in which structures appear that are vastly more complex than the rest of biological phenomena. This way the “humanities”, social sciences etc. appear in biology.

And so on.

To me it seems that the relation between physics and math is precisely of this sort. While math is not an empirical science, saying so is analogous to saying that chemistry is not bioscience: while true, the conclusion “so, they are different” drawn from it somehow misses the point.

Posted by: Urs Schreiber on June 30, 2008 12:08 PM | Permalink | Reply to this

Re: Michael Polanyi and Personal Knowledge

This sounds quite Polanyian. But I wonder how far you would follow him in his ideas on hierarchical levels and boundary conditions.

There is something limit-like, in the category theoretic sense, about the passage from subsidiary awareness to focal awareness.

Posted by: David Corfield on June 30, 2008 12:31 PM | Permalink | Reply to this

Re: Michael Polanyi and Personal Knowledge

But I wonder how far you would follow him in his ideas on hierarchical levels and boundary conditions.

Okay, I am reading…

Hm, this statement:

A machine, for example, cannot be explained in terms of physics and chemistry. Machines can go wrong and break down - something that does not happen to laws of physics and chemistry.

looks logically flawed to me.

There may be machines and, more importantly, there are natural phenomena, which we still cannot fully explain on the grounds of the theories which in principle we know ought to be sufficient for an explanation. (We can’t even properly derive bound states of nucleons (protons, neutrons) from QFT at the moment. How is that for a reductionistic world view?) But that’s not about machines breaking down and laws continuing to hold.

Then…

No inanimate object is ever fully determined by the laws of physics and chemistry.

That could be. But we don’t really know.

Then…

There are always some initial conditions necessary in order to have any system at all.

I suppose he means boundary conditions here.

Then…

The higher principle of structure controls the lower principles of physics and chemistry.

It’s a bit vague, but I think I’d agree with that: even a full reductionistic understanding (for instance the infamous “theory of everything”) may for practical purposes be insufficient for deriving all its consequences (even if in principle they all follow).

Maybe intersting in this context the little school of contemporary natural philosophers around Stephen Wolfram: they take this as their starting point of understanding nature: for them, even the reductionistic rules which have been found to govern the world are supposed to arise as emergent phenomena from very elementary rules, the emergence so untractable that they focus their research on just eyeballing computer output generated by simple automata, hoping to find the ultimate emergent phenomenon.

Skipping over the analogy with human speech…

The next step is to note that living beings are possessed of intelligence, another supervening principle,

Maybe I’d use “conciousness” instead of “intelligence” here. What is conciousness? A reductionistic “theory of everything” will hardly ever be of help for answering this (nor for making the question more well defined, which might be the more difficult problem).

Then:

Unbridled detailing, the ideal advocated by Laplace and his modern followers, not only destroys our knowledge of things we most want to know; it clouds our understanding of elementary perception - our first contact with the world of inanimate matter and of living beings and our initial act of self-transcendence.

This sounds like something I’d agree with. Somehow.

Then:

Against the self-destructive commitment to ultimate lucidity, I propose the theory of tacit knowing.

I understand that he is referring to a theory developed elsewhere here which I haven’t seen. But for me at this point it is mysterious why from the noticing emergent phenomena in nature and the difficutly of deriving them from the proverbial reductionistic laws leads him to talk about “tacit knowledge”.

But I have to run now to catch my train. Maybe more later.

Posted by: Urs Schreiber on June 30, 2008 7:16 PM | Permalink | Reply to this

Re: Michael Polanyi and Personal Knowledge

Perhaps I shouldn’t have started this discussion knowing I’ll only have the chance to snatch the briefest moment over the next three days.

But I shall return.

Posted by: David Corfield on July 1, 2008 9:41 AM | Permalink | Reply to this

Re: Michael Polanyi and Personal Knowledge

Going off on a bit of a tangent, Urs wrote

It’s a bit vague, but I think I’d agree with that: even a full reductionistic understanding (for instance the infamous theory of everything) may for practical purposes be insufficient for deriving all its consequences (even if in principle they all follow).

Maybe intersting in this context the little school of contemporary natural philosophers around Stephen Wolfram: they take this as their starting point of understanding nature: for them, even the reductionistic rules which have been found to govern the world are supposed to arise as emergent phenomena from very elementary rules

I can’t claim to be totally sure on this, but I think the Wolfram-ite philosophy is almost the reverse: any set of simple rules which isn’t trivial is supposed to be capable of generating essentially the same complexity, so if you’re interested in a behaviour at some level then it’s not particularly worthwhile to try and find the exact rules for all the lower levels and calculate the consequences of them exactly; you may as well try various simple rules and, in this world view, you should find many simple models which exhibit the phenomena you want to study.

It’s a view that I can see as at least possibly interesting from a scientific point where you are happy to “just” “understand” things (say, the kind of rules and conditions that give rise to hurricanes) but I’ve not seen how this translates into the cases where you don’t just want to understand but you actually want to engage in precise prediction (what would I do if I want to predict the weather in the Gulf of Mexico over the next couple of months?).

Posted by: bane on July 3, 2008 12:01 PM | Permalink | Reply to this

Re: Michael Polanyi and Personal Knowledge

Oops: The lead-in should read “I’ll go off on a bit of a tangent, commenting on what Urs wrote:”

Posted by: bane on July 3, 2008 12:12 PM | Permalink | Reply to this

Re: Michael Polanyi and Personal Knowledge

Maybe intersting in this context the little school of contemporary natural philosophers around Stephen Wolfram: they take this as their starting point of understanding nature: for them, even the reductionistic rules which have been found to govern the world are supposed to arise as emergent phenomena from very elementary rules

I can’t claim to be totally sure on this, but I think the Wolfram-ite philosophy is almost the reverse: any set of simple rules which isn’t trivial is supposed to be capable of generating essentially the same complexity, so if you’re interested in a behaviour at some level then it’s not particularly worthwhile to try and find the exact rules for all the lower levels and calculate the consequences of them exactly; you may as well try various simple rules and, in this world view, you should find many simple models which exhibit the phenomena you want to study.

I have certainly seen various Wolfram-ites, including Wolfram himself, who were hoping to find finite automata based on simple rules whose output displays effects that can be interpreted as particle scattering or the like.

Posted by: Urs Schreiber on July 4, 2008 10:50 AM | Permalink | Reply to this

Wolfram’s question; Re: Michael Polanyi and Personal Knowledge

Wolfram, when I told him that my teenaged son was earning his law degree to do Intellectual Property, asked who owns an algorithm data-mined from the set of all possible algorithms? I think that he was referring to algorithms, with simple rules, found by such data mining of the space of all possible algorithms, rather than reverse-engineered or conventionally engineered.

Posted by: Jonathan Vos Post on July 4, 2008 12:53 PM | Permalink | Reply to this

Re: Wolfram’s question; Re: Michael Polanyi and Personal Knowledge

Yes, that’s what they are doing at least in part of the “New-kind-of-science”-research: scan the space of all finite automata for one that produces output which looks like it is processing a universe such as ours – in some sense.

Suppose such a finite automaton exists which “processes our universe” (I can’t quite believe it does, but can’t exclude it either), then current high energy physics (QFT, gravity, standard model, etc.) would be to this algorithm as biology is to chemistry.

Posted by: Urs Schreiber on July 4, 2008 2:59 PM | Permalink | Reply to this

Re: Michael Polanyi and Personal Knowledge

There are certainly some who hope to find rules for particle physics. I was going more by the “natural philosophy” label (which to me means understanding anything that has a physical manifestation whether population biology or quantum mechanics), and just making the point that they don’t “not use lower-level theories” because those theories are computationally intractable since the viewpoint is that “everything is intractable” but rather the fact that rules set X “really is” the low level rules of say, particle physics, doesn’t mean it’s a particularly better way to study higher level phenomena than looking for new rules that give rise the phenomena you’re investigating. (In particular, their newly created rules are likely to be as undecidable/turing-complete/etc as the “true low level rules”.)

Posted by: bane on July 4, 2008 3:27 PM | Permalink | Reply to this

Einstein on Stratification; Re: Michael Polanyi and Personal Knowledge

Albert Winstein spoke of this, minus the Albert Einstein spoke of this, minus the Turing concept (pity that he didn’t make more use of his friendship with Godel):

Stratification of the Scientific System. The aim of science is, on the one hand, a comprehension, as COMPLETE as possible, of the connection between the sense experiences in their totality, and, on the other hand, the accomplishment of this aim BY THE USE OF A MINIMUM OF PRIMARY CONCEPTS AND RELATIONS. (Seeking, as far as possible, logical unity in the world picture, i.e. paucity in logical elements.)

Science uses the totality of the primary concepts, i.e. concepts directly connected with sense experiences, and propositions connecting them. In its first stage of development, science does not contain anything else. Our everyday thinking is satisfied on the whole with this level. Such a state of affairs cannot, however, satisfy a spirit which is really scientifically minded; because the totality of concepts and relations obtained in this manner is utterly lacking in logical unity. In order to supplement this deficiency, one invents a system poorer in concepts and relations, a system retaining the primary concepts and relations of the “first layer” as logically derived concepts and relations. This new “secondary system” pays for its higher logical unity by having elementary concepts (concepts of the second layer) which are no longer directly connected with complexes of sense experiences. Further striving for logical unity brings us to a tertiary system, still poorer in concepts and relations, for the deduction of the concepts and relations of the secondary (and so indirectly of the primary) layer. Thus the story goes on until we have arrived at a system of the greatest conceivable unity, and of the greatest poverty of concepts of the logical foundations, which is still compatible with the observations made by our senses. We do not know whether or not this ambition will ever result in a definitive system. If one is asked for his opinion, he is inclined to answer no. While wrestling with the problems, however, one will never give up the hope that this greatest of aims can really be attained to a very high degree.

An adherent to the theory of abstraction or induction might call our layers “degrees of abstraction”; but I do not consider it justifiable to veil the logical independence of the concepts from the sense experiences. The relation is not analogous to that of soup to beef but rather of check number to overcoat.

[Albert Einstein, “Physics and Reality”, J. Franklin Inst., Vol, 221, No. 3, March 1936]

Posted by: Jonathan Vos Post on July 4, 2008 6:24 PM | Permalink | Reply to this

Re: Michael Polanyi and Personal Knowledge

There never seems enough time to chat these days.

Polanyi makes a distinction between two situations involving boundary conditions.

First, test tube-like ones where you’re not interested in the boundary, but rather the operation of the lower level principles governing the contained material.

Second, machine-like ones where you’re interested in the boundary conditions, changes in which ride on the back of the physical substrate.

Polanyi’s idea was that in the machine case there’s no way to understand its design just in terms of the substrate:

…a machine can be smashed and the laws of physics and chemistry will go on operating unfailingly in the parts remaining after the machine ceases to exist. Engineering principles create the structure of the machine which harnesses the laws of physics and chemistry for the purposes the machine is designed to serve. Physics and chemistry cannot reveal the practical principles of design or co-ordination which are the structure of the machine.

Posted by: David Corfield on July 9, 2008 9:57 PM | Permalink | Reply to this

emergent phenomena Re: Michael Polanyi and Personal Knowledge

I agree with Urs that “emergent phenomena” in the Theory of Complex Systems are a refutation of the traditional hierachy of sciences in Reductionism.

In this sense, Polanyi anticipated the rise of a new methodology or paradigm or field of enquiry, which cuts across the old discipline boundaries, and goes far beyond Chaos Theory as a refutation of mechanistic determinism in dynamic systems.

I differ slightly with Urs in his statement tha Mathematics is not empirical. Although I agree that Mathematics is traditionally axiomatic as opposed to the experimental nature of sciences, I mentioned “Experimental Mathematics” for a reason. It has notable practioners, such as Borwein, its own journal, its own conferences, and presents a challenge to some Philosophies of Mathematics.

Posted by: Jonathan Vos Post on July 1, 2008 4:43 PM | Permalink | Reply to this

Re: emergent phenomena Re: Michael Polanyi and Personal Knowledge

I agree with Urs that “emergent phenomena” in the Theory of Complex Systems are a refutation of the traditional hierachy of sciences in Reductionism.

By the way, I was arguing that at each step of the chain

Math \to Physics \to Chemistry \to Biology \to Humanities(?) \to \cdots

attention is focused on a subset of the predecessor in which however a new level of complexity emerges which in principle but in practice only with difficulty may be entirely deduced.

I suppose this is uncontroversial and commonplace for arrows except the first one. My point was that to me the first arrow seems to be of the same general nature.

Posted by: Urs Schreiber on July 2, 2008 8:11 AM | Permalink | Reply to this

Tegmark and Egan; Re: emergent phenomena Re: Michael Polanyi and Personal Knowledge

Other attacks on the hierarchy of sciences are common in the literature.

Tegmark has claimed, as another thread here discussed, that the Physical Universe is IN FACT a Mathematical Structure as such. That chages the first arrow in your diagram.

However, as we’ve discussed in other threads, Mathematics does not spring from the null set free of charge, nor spontaneously. Nor from the brow of Zeus (as did Athena). And the Muse of Math is Urania, in any case, daughter of Mnemosyne, Urania also being the muse of Astronomy. That’s why Cosmology is big in this blog;) By the way, Michael Salamon took me and my wife to dinner and tells that he’s been put in charge of ALL cosmology missions at NASA HQ.

Mathematics TO WE HUMANS is possible because of evolution that selected for certain neurological and sensory and kinaesthetic capabilities.

Mathematical notation, as with written alphabets, may result from repurposing of visual and neural capabilities for abstracting the graphs of horizon, trails, vertical trees, and the like. The theory has been articulated that Math is based on how we physically manipulate objects and our own bodies.

Other intelligences, with different biology, may have VERY different Mathematics, although it may have natural transformations between a subset of theirs and a subset of ours.

Other universes, with different physical laws, may also allow elegant embeddings of other mathematics.

We don’t know with 100% certainty that the same Math applies to all parts of OUR universe. Greg Egan has written brilliantly on this in Luminous and other fictions.

Posted by: Jonathan Vos Post on July 2, 2008 7:33 PM | Permalink | Reply to this

Whither Mathematics; Re: Tegmark and Egan; Re: emergent phenomena Re: Michael Polanyi and Personal Knowledge

Egan’s fiction is plausible, because Kurt Godel: “established that the consistency of arithmetic was not provable…. It is, in fact, logically possible that Peano arithmetic is logically inconsistent…. Perhaps the shortest proof of an inconsistency in Peano arithmetic is one hundred million pages long. If we were never led into a contradiction, would the inconsistency matter? We could continue to prove theorems and derive interesting interconnections between ideas without ever suspecting the awful truth. Such a situation need not imply that our efforts were worthless….. A famous book of Imre Lakatos is a celebration of the ability of mathematicians to respond to counterexamples to a sequence of flawed statements in Euler’s theorem…. Interesting mathematics is remarkably tolerant of changes in the axiomatic framework, and can often be rescued from technical errors…. The twentieth century provided both of these conditions [mathematics that could only be validated at a collective level; ever more sophisticated computer software] for the decisive and irreversible change in the nature of mathematical research. Pure mathematics will remain more reliable than other forms of knowledge, but its claim to a unique status will no longer be sustainable…. We finally ask if there are further crises to be faced [beyond Godel; computer-assisted proofs too complex for individual human comprehensibility; intrinsic uncertainty of global correctness]. One possibility is the discovery of a contradiction in a mathematical argument whose complexity is beyond any yet contemplated. One might imagine that the contradiction is the result of a mistake that is too deep for us to be able to locate it, even with the aid of computers…. By 2075 many fields of mathematics will depend on theorems that no mathematician could fully understand whether individually or collectively…. the unique status of mathematician entities [Platonist Realist as with Penrose, or anti-realist as with Paul Cohen] will no longer seem relevant.”

“Whither Mathematics?”, Brian Davies, Notices of the AMS, Vol 52, No. 11, Dec 2005, pp.1350-1356]

Posted by: Jonathan Vos Post on July 4, 2008 9:27 PM | Permalink | Reply to this

Re: emergent phenomena Re: Michael Polanyi and Personal Knowledge

I’m not sure if this really constitutes disagreement with Urs’s claim about the relationship between math and physics, but I rather dislike the whole idea of the first step in the hierarchy

Math \to Physics \to Chemistry \to Biology \to

The reason is that this can easily give the impression to those outside science that mathematics used in science only explicitly appears in connection with physics. So one might think (for example) that biologists can only hope to use mathematics insofar as they can reduce the biological phenomena they study to basic processes of physics. Of course this impression bears no resemblance to mathematical biology in practice.

Posted by: Mark Meckes on July 18, 2008 6:01 PM | Permalink | Reply to this

Rota, Sturmfels, Bio/Math; Re: emergent phenomena Re: Michael Polanyi and Personal Knowledge

As someone who’s published more Mathematical Biology than Mathematical Physics, I very much agree with Mark Meckes.

“Can Biology Lead to New Theorems”, by Bernd Sturmfels, a feature article available from the Clay Institute web site, says in part:

“… some [mathematicians] secretly hope that this ‘biology fad’ will simply go away soon…. [has] anything really changed since Gian-Carlo Rota wrote… ‘The lack of real contact between mathematics and biology is either a tragedy, a scandal, or a challenge, it is hard to decide which’… [an] optimistic vision is expressed succinctly in the title of J.E. Cohen’s article: ‘Mathematics is biology’s next microscope, only better; biology is mathematics’ next physics, only better.’”

Sturmfels asks “will a theoretical biologist ever win a Fields medal?” He then gives some very pretty theorems from Biology, which I’ve also mentioned on another n-Category cafe thread.

Posted by: Jonathan Vos Post on July 18, 2008 11:30 PM | Permalink | Reply to this
Read the post Hierarchy and Emergence
Weblog: The n-Category Café
Excerpt: Emergence in hierarchies
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