The different kinds of logic
Mathematics is a good deal like the logic of propositions in one sense, in that it doesn't deal directly with the real world; but there are important differences between the way mathematics does things and the way formal logic (the logic of statements) does--which is why, as I said, it is not a good idea to model formal logic on mathematics. And, like formal logic, I think modern mathematics has also got itself into difficulties, in this case in the area of infinite sets, and so I am going to make a few critical comments; because I think the difficulties are philosophical, not mathematical, and it doesn't follow that good mathematicians know all about how mathematical thinking works, any more than it follows that good drivers know how cars work.
One reason mathematics is thought to be the same as logic is that it isn't an empirical science. If it was once thought that 5 + 7 = 12 was something rooted in the nature of things, it is now realized (correctly, I think) that the roots are considerably more tenuous than we thought they were. There are number systems (such as the hexadecimal system used in computers) in which 9 + 9 = 12 (because 12 in this system means "one 'sixteen' and two units). But that, of course, is a quibble of at what digit you choose to have the Arabic form of numbering system repeat.
However, there are different geometries from Euclid's today, which work quite well in the real world, thank you; and so Euclid's "axioms" are not axioms in the sense he thought of them: universal truths about how figures "really are."Next