#abstraction #mathematics #right tool for the job #theory #philosophy
epoch: 1259508138
In 20th century philosophy we may observe two main perspectives or views onto mathematics. These are alife even in our days. The first is to damn mathematics as a symptom of what Heidegger called “Seinsvergessenheit”. The other is to declare mathematics or mathematical logic as the only valid philosophy. There are only a few in the middle. One was Gotthard Günther and another seems to be Alain Badiou. But a closer look reveals Badiou as part of both extremes and not at all in the middle. He declares mathematics as the only valid ontology and on the other hand brings up philosophy as a theory of the “Ereignis”, a follower of Heidegger again. Gotthard Günther in contrast—also developing ideas of Heidegger—used mathematical or formal methods as a tool to speak about what natural language cannot say. In fact he was not in the middle of the extremes but beyond both since he rejected a wrong alternative.
Exactly that’s what mathematics is: it’s a tool, a language. In more detail, it’s a multitude of languages often saying the same in a different or “dual” way, just like natural languages do. A choice of a mathematical language aims to say something in a proper way. Physicists know about the fact that natural language is limited. To have a talk on quantum field theory without a fair amount of mathematics is hardly possible. Nevertheless, physicists usually don’t care that much on mathematical rigor or proofs. They formulate theories to make predictions of observations in the physical world. It’s similar with a computer scientist. All he or she wants is to automatically reproduce certain reductions of complexity in an efficient way. Mathematics is a tool to find such ways and to speak about features like optimality and completeness.
Even the mathematician does hardly more instead of refining the tools. Proofs ensure that the tool does not break. The most fundamental theories are not to recreate the world from scratch but to ensure the house does not collapse in the near future. This is what Daniel Goldblatt might have had in mind saying: “A Foundational system serves not so much to pop up the house of mathematics as to clarify the principles and methods by which the house was built in the first place”, (Topoi - The Categorial Analysis of Logic, p.14).
But, is really everything a tool? Is a toy a tool? What about a work of art? Here, we should not talk about the fundamental meaning of “tool” as an abstraction of even language or gesture itself ending with an absolute notion of “everything is a tool” or eventually “being is toolness” or something like that (could be interesting though). Anyway, playing with toys or enjoying art may be considered different. Sure, there is no better way to learn something than to play around with it. And there is no way to prove oneself better than creating art just to do it. But what we thought about here was less, just to have a good tool for a job—and mathematics is more often than not a proper one even in philosophy.
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