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*To*: "fom@math.psu.edu" <fom@math.psu.edu>, Jan Mycielski <jmyciel@euclid.Colorado.EDU>, Vladimir Sazonov <sazonov@informatik.uni-siegen.de>*Subject*: FOM: GCH for some cardinal nos.*From*: Vladimir Sazonov <sazonov@informatik.uni-siegen.de>*Date*: Sat, 04 Dec 1999 19:23:26 +0100*Sender*: owner-fom@math.psu.edu

Dear Professor Mycielski, I like very much your posting to FOM, especially your notes on rejecting Platonism and how you do this in the framework of set theory. I completely agree that > mathematics is a > human construction and not a description of an ideal world independent of > humanity. A construction which is physical (electrochemical processes in > brains, computer computations, and notes on paper) and is as real as other > physical objects made by people and machines. Further, I would like to somewhat reformulate the next phrase: Thus, in a real enough sense, mathematicians are very much *like* engineers, architects, painters or sculptors. Here mentioning engineers is particularly appropriate. For example, they construct air planes which strengthen our ability to move in the space. Mathematicians create somewhat different "devices" (like rules of multiplication of decimal numbers, rules of calculating derivatives or integrals in Analysis, logical modus ponens and reductio ad absurdum rules, etc.) which crucially strengthen our thought, our ability to describe and understand the real world. These are not rules *of* thought, but, rather, rules or levers *for* thought. In this sense mathematics is just a special kind of engineering. In this sense computers, as levers for thought, also may be used in (or belong to) mathematics. The above seems is in coherence with your views. But I cannot agree that > mathematicians are *no more* [my emphasis, - VS] formalists than engineers, architects, > painters or sculptors. What is then the crucial difference between mathematics and other activities? Mathematical rules are formal (rigorous), unlike the rules which use, say, sculptors in their work. Even lawyers use not so formal rules as mathematicians, and they hardly could even hope on the mathematical level of rigor. (Chess game has formal rules of the same rigor as in Math., but they hardly could be called levers for thought.) Formal rules is a *material of which mathematics is made*. Painters or sculptors use different materials. Actually, mathematicians need not always reach the fullest formality. But they always (at least since Euclid) had some ideal of what is rigorous rule or a proof. Each case when such an ideal was temporary lost was a drama. Contemporary logic gives good explication of this rigor. (But it could be discussed whether this rigor/formality is really fullest possible one.) Only due to formality the rules of mathematics and logic are so powerful. Formal - therefore we can use them mechanically and repeatedly, thinking thereby on more high level entities. (A specific property of the material, like that of metal for making a knife.) I have nothing against the term "rationalism". However, it seems cannot serve as replacement for "formalism". I do not understand why you do not want to be called a formalist. If your position is not (rational, reasonable) formalism, what is then formalism? The fact that formalism was often used in foundations of mathematics like a shameful brand does not mean that it is so bad. It seems we should simply recover the proper meaning of this word. Where from it follows that formalism deals with formal systems without any respect to their meaning? Say, engineers, in principle, also can build anything meaningless/useless by using the same rules and materials by which they construct a plane. So what? Best wishes, Vladimir Sazonov

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