## Experimental mathematics as synthetic aposteriori knowledge

Activity: Talk or presentation types › Lecture and oral contribution

Mikkel Willum Johansen - Lecturer

- Department of Science Education
- History and Philosophy of Science and Science Studies

Traditionally, mathematics has been considered to be a priori knowledge.

However, any kind of knowledge that in an essential way relies on a computer experiment must be considered to be aposteriori. Thus, from a traditional point of view (see eg. Tymoczko, 1979) the advent of computer experimentation in mathematics has put us in a dilemma: Either we must reject the idea that computer experimentation can give us mathematical knowledge proper or we must reject the idea that mathematical knowledge is a priori - and consequently we cannot explain what sets mathematics apart form

(other) empirical sciences.

In my view this dilemma is false. There is no viable definition of a priori that makes all of traditional mathematics come out as a priori. Thus, the introduction of computer experimentation has changed nothing; we gave up the apriority of mathematics long ago, only we have not realized it yet. On the other hand, this does not mean that the aposteriori knowledge produced in mathematics is qualitatively similar to the aposteriori knowledge produced by the natural sciences. In my view, the knowledge produced by the sciences is synthetic knowledge, while the aposteriori knowledge produced in mathematics is analytic knowledge. By categorizing parts of our mathematical knowledge as analytic aposteriori we can accept computer experimentation as parts of mathematics proper, and still explain why mathematical knowledge is different form knowledge produced by the natural sciences. In other words I will argue, that the dilemma introduced by philosophers such as Thomas Tymoczko can solved by developing an adequate understanding of the epistemological status of mathematical knowledge.

However, any kind of knowledge that in an essential way relies on a computer experiment must be considered to be aposteriori. Thus, from a traditional point of view (see eg. Tymoczko, 1979) the advent of computer experimentation in mathematics has put us in a dilemma: Either we must reject the idea that computer experimentation can give us mathematical knowledge proper or we must reject the idea that mathematical knowledge is a priori - and consequently we cannot explain what sets mathematics apart form

(other) empirical sciences.

In my view this dilemma is false. There is no viable definition of a priori that makes all of traditional mathematics come out as a priori. Thus, the introduction of computer experimentation has changed nothing; we gave up the apriority of mathematics long ago, only we have not realized it yet. On the other hand, this does not mean that the aposteriori knowledge produced in mathematics is qualitatively similar to the aposteriori knowledge produced by the natural sciences. In my view, the knowledge produced by the sciences is synthetic knowledge, while the aposteriori knowledge produced in mathematics is analytic knowledge. By categorizing parts of our mathematical knowledge as analytic aposteriori we can accept computer experimentation as parts of mathematics proper, and still explain why mathematical knowledge is different form knowledge produced by the natural sciences. In other words I will argue, that the dilemma introduced by philosophers such as Thomas Tymoczko can solved by developing an adequate understanding of the epistemological status of mathematical knowledge.

16 Nov 2012

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