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*To*: fom@math.psu.edu, holmes@catseye.idbsu.edu*Subject*: FOM: role of formalization in f.o.m.; coordinate systems*From*: Stephen G Simpson <simpson@math.psu.edu>*Date*: Tue, 1 Jun 1999 15:41:34 -0400 (EDT)*In-Reply-To*: <199906011757.LAA08170@catseye.idbsu.edu>*Organization*: Department of Mathematics, Pennsylvania State University*References*: <199906011757.LAA08170@catseye.idbsu.edu>*Reply-To*: simpson@math.psu.edu*Sender*: owner-fom@math.psu.edu

Holmes 1 Jun 1999 11:57:21 > I suggest that Simpson should carefully consider the possibility > that he is misunderstanding Conway's intentions But obviously I have considered it. I don't think I am misunderstanding Conway. > I see no "anti-foundational views" in this passage; Both Holmes and Shipman accuse me of accusing Conway of anti-foundationalism, but I never did so. I did however use the term *anti-formalization* to describe Conway's view that formalization is irrelevant. I still think that term is consistent with and justified by Conway's own remarks. One of Conway's relevant remarks is: It seems to us, however, that mathematics has now reached the stage where formalisation within some particular axiomatic set theory is irrelevant, even for foundational studies. The surrounding passage is reproduced in full in my posting of 24 May 1999 19:25:58. It's too bad Conway never took the trouble to clarify or amplify his anti-formalization remarks, so that we might understand their context and meaning in Conway's mind. I think it was somewhat irresponsible of Conway to make his anti-formalization remarks without a full, detailed explanation. I don't think Holmes and Shipman realize how much damage this kind of irresponsibility can cause. I think their attitude toward Conway is too charitable. Perhaps their enjoyment of the mathematical content of Conway's book has blinded them to the poor quality of Conway's foundational remarks. > neither, I think, do many other respondents in the discussion thus > far. Shipman 19 May 1999 14:25:44 says he agrees with me that Conway should not have said ``even for foundational studies''. But he offers no explanation of why Conway *did* say that. Neither does Holmes. > I think that Conway can be understood as objecting to views which > regarded some specific system (say ZFC) as the indispensible core > of foundations of mathematics: of course, Simpson either holds or > appears to hold views of exactly this kind ... I don't hold views of this kind. I do however hold that formalization is an essential part of modern f.o.m. research. Does anyone dispute this? > There is nothing in this which suggests that Conway is opposed to > formalization per se or does not recognize that it is important; On the contrary, Conway's remark that ``formalisation within some particular axiomatic set theory is irrelevant, even for foundational studies'' tends to show that Conway thinks formalization is irrelevant, even for foundational studies. The truth of the matter is that formalization is *very relevant*, especially for foundational studies, as f.o.m. professionals well know. Therefore, I characterize Conway's view as one of opposition to formalization. I don't know why Holmes thinks this is unreasonable on my part. > Using an analogy which Conway makes in his discussion in ONAG, > formalizations can be regarded as analogous to coordinate systems. In my opinion this is a very bad analogy. It is inappropriate and misleading. In geometry, there is a well known distinction between analytic and synthetic geometry. Synthetic geometry goes back to the ancient Greeks and is coordinate-free, in the sense that coordinate systems are never mentioned or used. Analytic geometry on the other hand is a modern development and makes essential use of coordinate systems. Historically, analytic geometry grew out of synthetic geometry by the introduction of coordinate systems. Each of the two kinds of geometry can stand on its own, but there are also some interesting theorems such as Desargues' theorem which elucidate the close relationship between them. In f.o.m., formalization of one kind or another is an important part of the fabric of virtually all ancient and modern f.o.m. research. (The Elements of Euclid are an attempt formalize synthetic geometry in an axiomatic framework.) Indeed, formalization is of the essence in f.o.m. There is no ``formalization-free'' approach to f.o.m. The idea of ``formalization-free'' f.o.m. is vacuous. The idea of trying to ``liberate'' f.o.m. from formalization is absurd. Thus the Conway/Holmes analogy between formalization and coordinate systems breaks down completely. Conway bluffed about having a metatheorem which would provide a formalization-free approach to f.o.m. However, he never stated the metatheorem, and if he ever does state it, I bet it will involve formalization anyway. -- Steve PS. I can think of at least one serious line of research which might suggest (to some people) the existence a formalization-free approach to f.o.m. Namely, there is the well known and extremely important f.o.m. idea of classifying foundational formal theories into hierarchies according to relative interpretability or relative consistency strength. However, this line of research is not actually formalization-free, because the formal theories are still there. And I don't think this is what Conway had in mind, because he didn't mention anything like this, and he dismisses formalization as irrelevant. Indeed, he didn't refer to any f.o.m. research or literature at all.

**References**:**holmes**- FOM: misunderstandings

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