Quotes from roger (in blue
"For example, he claimed that 5+7=12 is synthetic a priori. Unfortunately this evidences a complete lack of understanding of math. … 5+7=12 is a consequence of a particular system of axioms and definitions - … - other systems are possible and have been developed." 2 = oo (concept formation)
5 = ooooo (concept formation)
7 = ooooo oo (concept formation)
10 = ooooo ooooo (concept formation)
12 = ooooo ooooo oo (concept formation)
12 = ooooo ooooo oo = ooooo + ooooo oo = 5 + 7 (synthetic a priori judgement)
If you think that the above is not necessarily (a priori) true, but only a result of arbitrarily chosen "system of axioms and definitions", then I cannot help you.
There is a huge progress from the axiomatic method of Euclid (and a formalistic interpretation of logics) to Kant's synthetic a priori judgement, and big step backwards from Kant to the neo-axiomatic method ala Hilbert & Russel, which essentially is the same as the old one of Euclid (and no better).
"Which is understandable enough, since at that time we hadn't developed the underpinning of mathematics as formal systems, nor was he aware of the possibilities of non-Euclidian geometry, etc. …"
Insofar as geometries
are purely formal systems, they are essentially only like games with arbitrary rules, and haven't a lot to do with our knowledge of the world.
If the concept curvature
of non-Euclidian geometries is more than a word without meaning (outside the game), then it presupposes even, straight, non-curved space (where parallel lines never meet).
"Okay, first of all, parallel lines do meet in 4 dimensional curved space - which is how our universe is shaped."
In the same way you could say, parallel lines on the surface of a sphere do meet, such as the meridians (lines of longitude) of a globe.
It is obvious that you can reduce Kant's epistemology to absurdity, if you call geodesic lines
Related posts of mine:
Finite time in an INFINITE space-time continuum?
Kant & counterrevolution & Einstein
By the way, Immanuel Kant was also one of the first who presented biological evolution as a consistent hypothesis (see 1
"People will argue that I'm not supporting my argument. Again, fine - true enough. I recall the endless …"
The same is valid for me.