Founding Arithmetic on geometry

In the past I found some fleeting references that some (Frege in his later years being one of them) tried to found arithmetic not on set-theory and logic but on geometry and logic.

Unfortunedly Frege didn't say a lot about this.(see the answer from Peter Smith below on a previous version on this question)
Are there any other articles or introductions on how to build arithmetic on geometry?

Old version

(So that you may understandPeter Smith 's answer better ) (in an earlier answer Peter Smith replied that it was not Logic that Frege tried to found on geometry, but only arithmetic)

In the past I found some fleeting references that some (Frege in his later years being one of them) tried to found logic not on set-theory but on geometry. Is there a good introduction to this subject?

Solution 1:

"Frege in his later years ... tried to found logic not on set-theory but on geometry."

Not so. Frege did change his mind in later years, but not about the nature of logic, but about the nature of numbers, and in particular about the status of the claim that no two natural numbers have the same successor (and hence, as we might put it, about the claim that there is an infinity of natural numbers).

Roughly speaking, Frege came to think that language is misleading when it seems to use expressions that denote numbers-qua-objects. Rather, we should cleave to the fundamental thought of the Grundlagen that in making numerical claims we are attributing properties to concepts (thus "Jupiter has four Galilean moons" attributes a certain property to the concept ... is a Galilean moon of Jupiter, namely the-property-of-having-four-instances). So we should consistently think of numbers as second-level concepts.

But this sabotages the proof in the Grundlagen (or the Grundgesetze) that we don't "run out of" numbers. To get an infinite sequence of numbers 0, 1, 2, ... we need an infinite sequence of concepts under which respectively 0, 1, 2, ... things fall. Frege had used the wonderfully cunning trick of considering in turn the concepts "... is not self-identical" under which zero things fall, which (he earlier thought) gave him the number-as-object zero, and hence the concept "... is identical to zero" under which one thing falls. Which gives him the number-as-object one, and hence the concept "... is identical to zero or one" under which two things fall. Which gives him the number-as-object two, and hence the concept "... is identical to zero or one or two" under which three things fall, and .... Well, you can see how the story goes. This gives him an infinity of natural numbers, but (to repeat) on the assumption that numbers are objects in their own right.

Late Frege (in the 1920s) seems to have abandoned that assumption, so can't run this proof. He needs some other infinity of objects to play with. In a famous passage he writes

I myself at one time held it to be possible to conquer the entire number domain, continuing along a purely logical path from the kindergarten-numbers; I have seen the mistake in this. I was right in thinking that you cannot do this if you take an empirical route ... for if it could be so based, we should have to reconcile ourselves to the brute fact of the series of whole numbers coming to an end, as we may one day have to reconcile ourselves to there being no stars above a certain size. But here surely the position is different: that the series of whole numbers should eventually come to an end is not just false: we find the idea absurd. So an a priori mode of cognition must be involved here. But this cognition does not have to flow from purely logical principles, as I originally assumed. There is the further possibility that it has a geometrical source.

The idea is that it is geometry, rather than the logical trickery, that might supply us with an infinity of objects (spatial points, perhaps). But then, with this additional input, we can perhaps run a version of the construction of arithmetic, but with numbers consistently treated as second-order concepts.

So yes, later Frege makes a hopeful appeal to geometry -- but hoping to give an a foundation of arithmetic on the basis of logic-plus-geometry (though he never worked out the details). He certainly isn't suggesting we found logic on geometry.

Solution 2:

Basically, you are searching for Euclid's Elements Book V which covers the abstract theory of ratio and proportion.

In modern setting, see David Hilbert, The Foundations of Geometry (1902) : §15. AN ALGEBRA OF SEGMENTS.

All the development of geometry is reviewed into Francis Borceux, An Axiomatic Approach to Geometry. Geometric Trilogy I (2014); see Chapter 8. Hilbert’s Axiomatization of the Plane.