Speakpigeon

Speakpigeon

Total Posts: 118

Joined 01-10-2017

14 January 2019 10:25

There is an important distinction to be made between two conceptions of logic as an empirical or an abstract science.

Mathematicians who worked on the conception of a method of logic in the 19th century, Frege in particular, were essentially and explicitly motivated by the idea that a proper method of formal logic would help improve the rigour of mathematical proofs, a particular concern at the time, between the two extremes of Abel and Weierstrass. This suggests a view of logic as essentially not arbitrary and therefore as *essentially empirical*.

And in effect, mathematicians working on a method of logic at the time had to rely on the only empirical evidence available to them, i.e. Aristotle’s syllogistic theory, plus what other people since had said on the subject, including other mathematicians, as well as their own personal intuition, as to what formulas could be accepted as logical truths, this in order to work out a method of logical calculus *they could use to improve rigour of proof*.

Today, *on the surface*, we seem to have a very different perspective, whereby logic is more often understood as essentially a mathematical object, like the set of Real numbers is, so that logic is thought of as being the methods of logic themselves that mathematicians have contrived since Frege. In this perspective, logic is no longer seen as an essentially empirical science, but as the motley collection of theories, *seen as arbitrary at least in principle*, that mathematicians are working on as objects of study rather than as methods they could use to improve the rigour of proofs.

Meanwhile, mathematicians themselves still essentially use and effectively rely on their own, intuitive, sense of logic to prove theorems, producing what can be described in effect as semi-formal proofs.

The few examples of formal logic being used to prove theorems today all rely on some variation of Gentzen’s “*natural*” method of proof (conceived between 1929 and 1935), which is essentially a modern generalisation of Aristotle, and a method which effectively relies on the crucial use of so-called rules of inference, which are formulas all essentially taken from the set of formulas long recognised as logical truths in the Aristotelian tradition, save a few exceptions.

So, in effect, all current practice of mathematical proof, be it intuitive or making use of theorem provers, like Isabel in Germany and Coq in France, still literally relies ultimately on the empirical evidence available to mathematicians that some logical truths are *evidently* true. Yet, the fundamentally empirical nature of the logic practised by mathematicians themselves, *today as always since Euclid*, is somewhat airbrushed out of the picture in favour of a more abstract notion of it.

Comments welcome.

EB