[FOM] Mathematics ***is*** formalising of our thought and intuition

Keith Brian Johnson joyfuloctopus at yahoo.com
Sat Jun 5 12:16:36 EDT 2010

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Isn't logic the formalization of what is thought to be correct or legitimate thought, with application to any possible objects, abstract or concrete?
Isn't theoretical mathematics the use of logic to describe patterns, with those patterns conceived abstractly--and therefore the formalization of the application of logic to patterns--and applied mathematics the use of logic to describe patterns, with those patterns instantiated concretely--i.e., ultimately the application of theoretical mathematics's formalized pattern analysis--although, of course, we may in fact do applied mathematics first and do theoretical mathematics second?

Mathematicians may certainly use their intuitions and unformalized thought to arrive at mathematical results, but isn't it precisely the role of foundational mathematics to demonstrate how those results can be derived from the application of logic to patterns, and the role of logic to formalize how we think about anything at all--i.e., to formalize the process of legitimately deriving conclusions from premisses?  Once that process has been formalized, then the results can be applied to the study of abstract patterns; and the results of the study of abstract patterns can then be applied to concretely instantiated patterns.  (Admittedly, this may be the reverse of how we initially learn about such things; but such is the nature of formalization--it follows intuition rather than preceding it.)

Thus, I take economists, political scientists, biologists, chemists, and physicists to be *using* mathematics and logic, *applying* them to concretely instantiated patterns, and thus to be species of applied mathematicians (when using mathematical thought processes; I do not claim that economists and political scientists always do so).
I take theoretical/pure mathematicians to be *using* logic in order to *formalize* our thought and intuitions about patterns in the abstract.
I take logicians to be *formalizing* our thought and intuitions themselves--to be formalizing what we take to be proper reasoning processes.

(Of course, someone whose job title is "physicist" might do theoretical/pure mathematics; someone whose job title is "economist" might do logic.  I am only claiming that a physicist qua physicist is only applying mathematics and logic, and that an economist qua economist is only applying mathematics and logic.)

Keith Brian Johnson

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