Where Chris Collins enters the fray

Chris sent me this longish response to some of what has appeared in the blog. In the hope of getting him to become a regularish participant in the ongoing discussions I here post his "Response to Norbert." I feel that he let me off lightly, actually. But this said, I think that I can still finds some points to disagree with. I will restrict these to the comments section and hand the floor over to him. Thx Chris.

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Response to Norbert

I read with interest Norbert’s recent post on formalization: “Formalization and Falsification in Generative Grammar”. Here I write some preliminary comments on his post.  I have not read other relevant posts in this sprawling blog, which I am only now learning how to navigate. So some of what I say may be redundant. 

For me the quote by Frege in the Begriffsschrift (pg. 6 of the book “Frege and Godel”) indicates what is important when he analogizes the “ideography” (basically first and second order predicate calculus) to a microscope: “But as soon as scientific goals demand great sharpness of resolution, the eye proves to be insufficient. The microscope, on the other hand, is perfectly suited to precisely such goals, but that is just why it is useless for all others.” Similarly, formalization in syntax is a tool that needs to be employed when needed. It not an absolute necessity and there are many ways of going about things (as I discuss below). By citing Frege, I am in no way claiming that we should aim at the same level of formalization that Frege did.

There is an important connection with the ideas of Rob Chametzky (posted by Norbert) in another place on this blog. As we have seen, Rob divides up theorizing into meta-theoretical, theoretical and analytical.  Analytical work, according to Chametzky is: “concerned with investigating the (phenomena of the) domain in question. It deploys and tests concepts and architecture developed in theoretical work, allowing for both understanding of the domain and sharpening of the theoretical concepts.” It is clear that more than 90% of all linguistics work (maybe 99%) is analytical, and that there is a paucity of true theoretical work.

A good example of analytical work would be Chomsky’s “On Wh-Movement”, which is one of the most beautiful and important papers in the field. Chomsky proposes the wh-diagnostics and relentlessly subjects a series of constructions to those diagnostics uncovering many interesting patterns and facts. The consequence that all these various constructions can be reduced to the single rule of “wh-movement” is a huge advanced, allowing one insight into UG. Ultimately, this paper lead to the Move-Alpha framework, and indirectly to Merge (the simplest and most general operation yet).

However, “On Wh-Movement” is what I would call “semi-formal”. It has semi-formal statements of various conditions and principles, and also lots of assumptions are left implicit. As a consequence it has the hallmark property of semi-formal work: there are no theorems and no proofs. Formalization is stating a theory clearly and formally enough that one can establish conclusively (i.e., with a proof) the relations between various aspects of the theory and between claims of the theory and claims of alternative theories.

Certainly, it would have been a waste of time to fully formalize “On Wh-Movement”. It would have expanded the text 10-20 fold at least, and added nothing. This is something that I think Pullum completely missed in his 1989 paper on formalization. The semi-formal nature of syntactic theory, also found in such classics as “Infinite Syntax” by Ross and “On Raising” by Postal, has led to a huge explosion of knowledge that people outside of linguistics/syntax cannot really understand (hence all the lame discussion out there on the internet and Facebook about what the real accomplishments of generative grammar have been), in part because syntacticians are not very good popularizers.

Theoretical work, according to Rob is:  “is concerned with developing and investigating primitives, derived concepts and architecture within a particular domain of inquiry.” There are many good examples of this kind of work in the minimalist literature. I would say Uriagereka’s original work on multi-spell-out qualifies and so does Epstein’s work on c-command, amongst others.

My feeling is that theoretical work (in Chametzky’s sense) is the natural place for formalization in linguistic theory. The reason is that it is possible, using formal assumptions to show clearly the relationship between various concepts, assumptions, operations and principles. For example, it should be possible to show, from formal work, that things like the NTC and Extension condition should really be thought of as theorems proved on the basis of assumptions about UG.  Since NTC and Extension condition are theorems, they can actually be eliminated from UG. And from this, one can wonder if that program can be extended to the full range of what syntacticians normally think about as constraints.

In this, I agree with Norbert who states: “It can lay bare what the conceptual dependencies between our basic concepts are.” Furthermore, as my previous paragraph makes clear, this mode of reasoning is particularly important for pushing the SMT forward. How can we know, with certainty, how some concept/principle/mechanism fits into the SMT? We can formalize and see if we can prove relations between our assumptions about the SMT and the various concepts/principles/mechanisms. Using the ruthless tools of definition, proof and theorem, we can gradually whittle away at UG, until we have the bare essence. I am sure that there are many surprises in store for us. Given the fundamental, abstract and subtle nature of the elements involved, such formalization is probably a necessity, if we want to avoid falling into a muddle of unclear conclusions.

A related reason for formalization (in addition to clearly stating/proving relationships between concepts and assumptions) is that it allows one to clarify murky areas. One of the biggest such areas nowadays is whether syntactic dependencies make use of chains, multi-dominance structures or something else entirely (maybe nothing else). Chomsky’s papers, including his recent ones, make references to chains at many points. But other recent work invokes multi-dominance. What are the differences and relations between these theories and are either of them really necessary? What assumptions about UG does multi-dominance or chains entail? I am afraid that without formalization it will be impossible to answer these questions. I am investigating these questions in my seminar this semester.

These questions about syntactic dependencies interact closely with TransferPF (Spell-Out) and TransferLF, which to my knowledge, have not only not been formalized but not even stated in an explicit manner. Investigating the question of whether multi-dominance, chains or some something else entirely (perhaps nothing else) is needed to model human language syntax will require a concomitant formalization of TransferPF and TransferLF, since these are the functions that make use of the structures formed by Merge.

Minimalist syntax calls for formalization in a way that previous syntactic theories did not. First, the nature of the basic operations is simple enough (e.g., Merge) to make formalization a real possibility. The baroque and varied nature of “transformations” in the “On Wh-Movement” framework and preceding work made the prospect for a full formalization more daunting.

Second, the nature of the concepts involved in minimalism, because of their simplicity and generality (e.g., copies, occurrences), are just too fundamental and subtle and abstract to resolve by talking through them in an informal or semi-formal way. With formalization we can hope to state things in such a way to make clear conceptual and empirical properties of the various proposals, and compare and evaluate them. In fact, I have recently being doing a lot of this with my colleagues, because only recently (by helping to write Collins and Stabler 2012) have I seen what the issues are.

So, in the spirit of Frege, formalization should be a tool for ordinary working syntacticians to clarify their ideas and examine them empirically and conceptually.

Formal Wear Part II: A Wider Wardrobe

Formal Wear Part II: A Wider Wardrobe

So can ‘formalization’ in the relevant sense (i.e. highlighting what’s important, including relevant consequences, while suppressing irrelevant detail, and sufficiently precise that someone else can use it to duplicate experiments or even carry out new ones)  sometimes be useful, serving as a kind of good hygiene regime to ‘clarify the import of our basic concepts’?  Certainly! Since the examples in the blog comments don’t seem to have strayed very far from a single-note refrain of weak generative capacity and the particular litmus test of ‘mild context-sensitivity,’ I thought it might be valuable to resurrect three concrete cases from the past that might otherwise go unnoticed, just to show how others have played the linguistic formalization game: (1) Howard Lasnik and Joe Kupin’s, A Restrictive Theory of Transformational Grammar (“a set theoretic formalization of a transformational theory in the spirit of Chomsky’s Logical Structure of Linguistic Theory”, 1977), to be posted here when I can get this link active; (2) Eric Ristad’s formalization and demonstration of the computational intractability of a series of linguistic theories: phonology (both segmental and autosegmental); the ‘original’ version of GPSG and the ‘revised’ GPSG of Gazdar, Klein, Pullum, and Sag (1985), here, here, and here; and then, in fact every modern linguistic theory, here; and (3) Sandiway Fong’s 1987-90 Prolog implementation of government-and-binding theory’s principles and parameters approach, that covered most of the examples in Lasnik and Uriagereka’s textbook, along with multiple languages (Japanese, Dutch, Korean, Bangla, German,…) here.  (There’s also my own 1984 demonstration here that “government-binding theory” (GB) grammars are semi-linear – i.e., like TAGs, they fall into the ‘sweet spot’ of mild context-sensitivity, here; but modesty forbids me from diving into it, and besides, it’s outdated, probably wrong, and just one more weak generative capacity result.) Outside of (1), I’d wager that not one linguist or computational linguist in a thousand knows about any of these results – but they should, if they’re interested at all in how formalization can help linguistic theory.  So let me march through each of them a bit, leaving the still-hungry (or bored) reader to follow-up on the details.

Here are the opening lines of Lasnik and Kupin (1977): “This is a paper on grammatical formalism…we are attempting to present a particular theory of syntax in a precise way…our theory is very restrictive…first, [because] the ‘best’ theory is the most falsifiable…and in the absence of strong evidence [otherwise] if that theory predicts the occurrence of fewer grammar-like formal objects than another theory, the former must be preferred….the second reason for positing a restrictive theory confronts the question of language acquisition” (p.173). L&K go on to show real ecological prescience: no trees were harmed in the making of their transformational movie! – because trees turn out to be merely a chalkboard-friendly, but not quite correct, graphical depiction of the relations one actually needs for the transformational substrate in LSLT, a set of strings, or Phrase Markers (PMs). As Howard puts it in his talk on the 50^th anniversary of the MIT Linguistics Dept. in 2012: “Chomsky’s theory was set theoretic, not graph theoretic, so no conversion to trees was necessary, or even relevant.”  I still don’t think most people even realize this. For instance, borrowing an example from Lasnik, the sentence “he left” would have the PM, {S, he left, he VP, he V, NP left, NP VP, S}, a representation of the fact that “he” is an NP; “he left” is an S; and so on. L&K formalize all this and more, reaping all the benefits formal hygiene advertises: by using an inductive definition instead of a generative one for PMs, L&K discovered that the PM definition is broader than necessary – the job of fixing all the ‘is-a’, relations in a sentence works just fine if one uses only reduced phrase markers (RPMs) – in our example, just the set {S, he VP, he V, NP left}, that is, all the elements of the original PM that have just a single nonterminal and any number of terminals, including 0. The reader should check that these suffice just as well as PMs in fixing all and only the “is-a” relationships of a sentence; e.g., given “he VP” and “he left”, one can conclude that “left” is a VP.  So this formalization has already told us: (1) the LSLT theory is too general, and can be restricted – so aiding learnability, as L&K note; and (2) we don’t need a phrase structure grammar at all, just transformational rules. Similar learnability considerations led L&K’s formalization to restrict transformations so that they were not marked as either optional or obligatory – that is to say, unordered transformational rules, unlike the complex “traffic rules” in both LSLT and Aspects. (See Howard Lasnik’s paper, “Restricting the theory of transformation grammar,” reprinted in his book, Essays on Restrictiveness and Learnability, 1990.) But then, as Howard notes, if you don’t need phrase structure rules, and all you need is transformations, what’s left? A linguistic theory where there is only one kind of structure building operation – an early version of minimalism! But wait, there’s still more. Formulating TG as juggling sets leads immediately to a satisfying account of some otherwise thorny problems – for one thing, it becomes easier to view coordination, quantifier ordering, and other ‘non tree-like’ parts of syntax as just the ‘spell out’ (linearization) of the set-union of RPMs (proposed by Grant Goodall in the 80s and implemented in 1983 by Sandiway Fong and myself in Prolog here, so another example of a precise, explicit, computable formulation).

OK, now what about Eric’s string of complexity results?  First, the obvious: evidently, there’s more to formalization than weak generative capacity.  To my mind, computational complexity results count as “formalization” just as much as weak generative capacity arguments, and ditto for any precise computational implementations. The litmus test for good models hangs on the “sufficiently precise” clause.  Second, what Eric showed goes far beyond the usual result that one or another linguistic theory has this or that complexity – e.g., that the languages generated by TAGs are efficiently parseable.  Rather, Eric showed something much more: that certain empirical properties about small parts of knowledge of language that everyone agrees on, embed certain problems that rise above the level of any one particular theory. By figuring out the computational complexity of such problems, we can draw conclusions about any linguistic theory that contains them, no matter what representation or algorithm we might consider.  (This just follows Marr’s prescription to consider problems in psychophysics, e.g., ‘stereopsis’ independently of theories, algorithms, and implementations.) For instance, suppose the problem is to determine the ‘obviation’ (non-coreference) relations in sentences such as, “Bill wanted John to introduce him,” what Eric calls the anaphora problem. If we can show that this computation is intractable, then this intractability infects all the rest of the language (or grammar) of which it is a part. There is no escape: if it were true that by considering all the rest of the language (or grammar) this problem became efficiently solvable, then Eric showed that this would imply that many known intractable problems (viz., those that are “NP-complete”) would also become efficiently solvable.  On the (widespread) assumption that P≠NP, this seems unlikely.   Further, as Eric notes, “this is true no matter how this [anaphora problem] is couched, whether in terms of constraints on a syntax relation of coindexing or linking, in terms of syntax or discourse, in terms of speaker-hearer intentions or other pragmatic considerations, or even in terms of a Montague-like compositional theory of semantic types. If the theory provides an empirically adequate description of the language user’s knowledge of utterances, then it will inherit the inalienable computational structure of that knowledge” (1990:112, Emph. added).

Note that this ‘intractability infection’ from part to whole stands in stark contrast to what happens with typical generative capacity results, where if we show that some particular construction, e.g., aⁿbⁿ is ‘complex’, e.g., strictly context-free instead of finite-state, then in general this complexity does not carry over into the full language (or grammar) – for instance, suppose aⁿbⁿ is a subset of a full language of any combination of a’s and b’s, a*b* – obviously just a finite-state language. Rather, in such cases one must also posit a set of mappings that strip the language, say English, down to just the particular construction in question, taking care that the mappings themselves do not introduce any ‘context-freeness’.  In my view, it is the ability to focus directly on a particular problem without having to worry about the rest of a language or grammar (or even the linguistic theory behind them) that makes complexity analysis such a powerful tool – a point that does not seem to have been fully appreciated.

So exactly what empirical bits about knowledge of language does Eric tackle? It’s hard to do justice to them all in just a short space, but the bottom line is they all they boil down to effects arising from agreement and ambiguity, which pop up in many places in human language.  Among these are facts about agreement and ambiguity  – “police police police” and all that – as well as facts about what we’ve already dubbed ‘obviation’ – non-co-reference, e.g., sorting out which pronouns can belong to which names in sentences like, “Before Bill, Tom and Jack were friends, he wanted him to introduce him to him”; head-head agreement, and so on.   All of these lead to computational intractability.  There’s a pattern here, that Eric comments on and I think is worth repeating, since I feel it’s one of the big downsides of formalization, and that’s the siren song of ‘mathematical purity’ – the (aesthetically gratifying) notion that human language really ought to be like physics, and really is a formal language.  I confess that I’m also strongly tempted by that song.

But as Eric remarks, the search for such mathematical purity has its drawbacks. His comment is worth quoting in full: “The pursuit of general mechanisms for linguistic theory – such as feature unification, the uniform local decomposition of linguistic relations, or co-indexing in Barriers – have repeatedly proven treacherous in the study of language. It distracts attention from the particular details of human language….General mechanisms have also invariably resulted in unnatural intractability, that is, intractability due to the general mechanisms of the theory rather than the particular structure of human language.  This is because no one mechanism has been able to model all the particular properties of human language unless it is the unrestricted mechanism. However, the unrestricted mechanism can also model unnatural properties, including computationally complex ones….In current syntactic theories, many types of agreement are used, including specifier-head, head-complement agreement (selection), head-head agreement, head-projection agreement, and various forms of chain agreement…when all these particular types of agreement are subsumed under one general mechanism, be it unification or co-indexing, unnatural forms of agreement invariably arise from interactions…. In a way these overgeneralizations reflect the mindset of formal language theory, which is to crudely equate structural complexity with syntactic form…. The remedy is, we must adopt the mindset of computational complexity theory, which is to equate structural complexity with computational resources. By limiting resources, we limit the number of possible rule interactions. The only way to satisfy these limits is to look for a more powerful class of linguistic constraints, that limit interactions among linguistic processes” (71-72. Emph. added).

So, third, though results like Ristad’s have often been dissed, to my mind they speak loud and clear.  And what they say is this: If you were somehow praying that linguistic theory alone would explain why human parsing is as fast as it seems, then it appears to me you’ve been going to the wrong church. Recall these hopeful words from 1979: that by restricting ourselves to grammars that generate only context-free languages “we would have the beginnings of an explanation for the obvious, but largely ignored fact that humans process the utterances they hear very rapidly.” Hopeful, yes; but also dead wrong. As far as I can make out, all current, descriptively adequate linguistic theories pose computationally intractable parsing problems. Yes, you read that right: all of them, from GPSG to HPSG, to LFG to non-projective dependency grammars, to TAGs and MCTAGs, to MCFGs, to, well, all of them.[1]  In other words: we’re all in the same complexity soup, all of us, together.  Now, I find that somewhat comforting, since so many aspects of modern life are, you know, alienating, and this one brings us all together under the same tent. More to say on this score in the blog on computational complexity.

Since this post has rambled on far too long already, perhaps it might be best to close with a point that Alex also raised about the necessity for mathematical arguments whenever one wants to establish some property about human language/grammar, e.g., that human grammars “have hierarchical structure” because, as Alex put it, “there is no way you can disprove a universal claim about grammars without proving something mathematical, because of this problem of universal quantification over grammars.” That’s well put, and bears reflection, but in such cases I find myself turning to the following rules for advice, which, after all, seems to have served us all pretty well:

“ Regula III. Qualitates corporum quæ intendi & remitti nequeunt, quæque corporibus omnibus competeunt in quibus experimenta instituere licet, pro qualitatibus corporum universorum habneda sunt.”

(“The qualities of bodies, which admit neither intension nor remission of degrees, and which are found to belong to all bodies within the reach of our experiments, are to be esteemed the universal  qualities of all bodies whatsoever.” Emph. added)

“Regula IV. In philosophia experimentali, propositiones ex phænomenis per inductionem collectæ, non obstantibus contrariis hypothesibus, pro veris aut accurate aut quamproxime haberi debent, donec alia occurrerint phænomena, per quæ aut accuratiores reddantur aut exceptionibus obnoxiæ.” (Translation left as an exercise for GoogleTranslate or the Reader.)

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[1] At this point, I imagine some of you are muttering to yourself: “but…but…but…what about my favorite theory?” Don’t you worry, you haven’t been forgotten. We’ll come back this in the upcoming blog on computational complexity. I’ll flag a warning now though: the words descriptively adequate are in there for good reason. So, that includes what I consider to be standard stuff, like scrambling and Condition B and quantifier scope. Now go back and read Eric’s results on obviation. And no, TAGs don’t escape: as soon as one has to pose the anaphora problem for them, one has to paste in add-ons to yield ‘multicomponent synchronous TAGs’ (Storochenko & Han, 2013), that, alas, lead one inexorably to intractability, as discussed in an excellent paper by Nesson et al. 2010, “Complexity, parsing, and factorization of tree-local multi-component tree-adjoining grammar,” in the Journal of the Association for Computational Linguistics. Their results have an interesting link to the complexity of Spell-out generally – but more about that in the upcoming blog.  Anyway, the bottom line is that I’ve seen no convincing escape hatch yet that works – not even that handy, all-purpose escape to semantics. And no, ‘concealed reference set computation,’ as suggested in some circles, doesn’t work either. Sorry.

Berwick Post: Formal Wear: Part I

I am just the blog tech here (something I find amusing given Bob's and my relative skill sets):

Formal Wear: Part I

While Fearless Leader (aka Norbert) was away from his computer, calculating how corks bounce off clay, he asked me to weigh in about this blog’s recent round-robin in the comments section among Alex Clark, David Pesetsky, and Norbert on the value of formalization in linguistics. Like Norbert, it should come as no surprise that I side with David here.  But as Norbert says, this same debate has popped up so often that it surely qualifies for a bit of historical perspective – some scientific success stories and some cautionary notes – echoing Norbert that “when done well formalization allows us to clarify the import of our basic concepts. It can lay bare what the conceptual dependencies between our basic concepts are.”  While the comments flying back and forth seem to have largely focused on just a narrow gang of the “usual suspects” – weak generative capacity and those pesky Swiss German and West Flemish speakers, and now, even Georgians – below I’ll describe several somewhat different but successful formalization blasts from the past. Getting old has one advantage: I can still actually remember these as if they were yesterday, far better than I can remember where I put my glasses.  And if there’s one thing these examples highlight, it’s another familiar hobby-horse: if you think that the world of linguistic formalization’s exhausted by weak generative capacity, mild context-sensitivity, and West Flemish, then look again – there’s more in heaven and earth than dreamed of in your philosophy.  As for the claim that formalization ought to be de rigeur for generative grammar, “just like” every other scientific field, as far as I can make out, it’s precisely the reverse: every other scientific field is “just like” generative grammar. 

So, why formalization? Alex runs through several very plausible reasons, from  “being precise” to “fully mathematically precise” before settling on a candidate that even I could vote for – we’ll come to what that is in a sec. But at first, Alex seems to equate ‘being precise’ with ‘mathematical formalization,’ all in the service of testability: “if your theory is precise, then you can use mathematical or computational methods to produce predictions.”  Well sure, sometimes.  But in truth the implication here really doesn’t work in either direction.  On the one hand, we have theories that pull out all the stops when it comes to mathematics, but still don’t quite reach the brass ring of testable predictions – string theory being the current poster child.  Before that, quantum physics preceded its formalization by Heisenberg, Dirac, and von Neumann by decades – and as Mario Livio remarks in his recent Brilliant Blunders, “More than 20% of Einstein’s original papers contain mistakes of some sort…[but often] the final result is still correct.  This is often the hallmark of great theorists: they are guided by intuition more than by formalism.” And then there are famous statements of this sort, that somehow have slipped by the “all must be formalized” censor: Nam tempus, spatium, locum et motum ut omnibus notissima non definio.” (“I do not define time, space, place and motion, as being well known to all.”) On the other hand, we have scientific theories of extraordinary precision, rivaling anything in physics, that aren’t ever mathematically formalized, and probably never will be, yet serve up a bumper crop of empirical tests and Nobel prizes year after year – modern biology and molecular biology standing front and center.[1] It’s perhaps worth remembering that what’s generally considered one of the greatest theories in all modern science, evolution by natural selection, was conceived by history’s most famous medical school dropout who later confessed in his autobiography that he “could not get past the very first steps in algebra” and “should, moreover, never have succeeded with…mathematics.”  If you believe that molecular biologists lie awake at night fretting over whether the cell’s intricate biomachinery choreographing the dance from DNA to protein must be cast as some set of axioms or equations – well, guess again. Biologists are a pragmatic lot. They lie awake at night thinking about how to get their experiments to work. Actually, in this respect biology is even worse off than linguistics: there are no laws[2] comparable to wh-island effects, let alone something sterling like F=ma. But wait, it gets worse (or better). As Francis Crick’s protégé Sydney Brenner will sing you loud and long, current molecular biology suffers from a severe lack of theories altogether, let alone formalized ones.  No matter; the gene jockeys just plough straight through all the formalism talk and still count on that regular airline flight to Stockholm in December.

In my opinion, Alex advances the winning candidate near the end of his volleys with David when he seemingly softens his mathematical stances and writes: “When I said formalisation in the last sentence I meant a proper mathematical model, rather than just providing a little bit of technical detail here and there. But no more than is used in any other branch of science.  I also don’t think that one necessarily needs to write a grammar for the whole language; but rather a mathematically precise model for a simplified or idealised system; but only for a part of the grammar.  That is how more or less all science works, and I don't think linguistics should be any different” [emphasis in the original].

Exactly! At so last we’ve arrived at a prescription very dear to my own heart: models. Models indeed loom large in science – the very heart and soul of the Galilean method, where the Master once proclaimed that he did not really understand anything in the physical world, like balls rolling down inclined planes, unless he was able “to build a machine that could reproduce its behavior,” and of course, more famously, in Il Saggiatore that “La filosofia è scritta in questo grandissimo libro… Egli è scritto in lingua matematica.” But there’s one final wrinkle: many perfectly fine scientific models aren’t mathematical equations – they can be anything from Feynman diagrams; to scaled-down versions of jet planes in wind tunnels; to computer simulations of ‘agents’ in SimCity and to computer programs generally; to the storyboard sequences of graphical ‘cartoons’ so popular with modern (molecular) biologists. To qualify as a good model, one must meet the dual demands of highlighting what’s important, including relevant consequences, while suppressing irrelevant detail, with sufficient precision that somebody else can use the model to replicate experiments or carry out new ones – which is to say, figure out whether to stick with one’s current theory ditch it to build a better one.  And that covers the waterfront.

At long last we’ve arrived at a prescription very dear to my own heart: models. Models indeed loom large in science – the very heart and soul of the Galilean method, where the Master once proclaimed that he did not really understand anything in the physical world, like balls rolling down inclined planes, unless he was able “to build a machine that could reproduce its behavior,” and of course, more famously, in Il Saggiatore that “La filosofia è scritta in questo grandissimo libro… Egli è scritto in lingua matematica.” But that’s one final wrinkle: there are many perfectly fine scientific models that aren’t mathematical equations – anything from Feynman diagrams, to scaled-down versions of jet planes in wind tunnels, to computer simulations of ‘agents’ in SimCity and to computer programs generally, to the storyboard sequences of graphical ‘cartoons’ so popular with modern (molecular) biologists.  The key requirements seem to be that a good model must meet the dual demands of highlighting what’s important, including relevant consequences, while suppressing irrelevant detail, and sufficiently precise that someone else can use it to duplicate experiments or even carry out new ones.  And that covers the waterfront. We’ll see how far in Part II.

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[1]I know of only a handful of well-intentioned, but failed, efforts to ‘formalize’ biology, one the famous program by Nicholas Rashevsky at the University of Chicago in the 1930s and another the much less well-known attempt by Eörs Szathmarthy in the 1980s to develop a computer programming formalization for biology grounded on the lambda calculus.

[2]Alas for Norbert’s obscure object of desire, there are actually good reasons to believe that we’re never going to find laws in biology like F=ma in physics, or what Feynman describes in his The Character of Physical Law, but taking up this point here would draw us a bit away from the main line of discussion.  Ask me about it.