Quora answer: Does mathematics help us understand our sense of logic or did our evolved sense of logic help us create mathematics?
This is an interesting and deep question.
There are actually three things to be concerned with:
- Logic as the core of Language
- Various possible foundations of Mathematics
- Mathematics proper.
The basic answer is that no one knows what the relations between these things are. Does language or numbers come first, and then what is the relation between the logical core of language and the foundations of mathematics, and finally what is the relation of the foundations to all forms of mathematics which seems to be expanding very rapidly over the last few centuries but almost exponentially in the last century.
I have written quite a bit about this question in response to Badiou’s Being and Event where he claims that Ontology is Set Theory. There are a lot of problems with this position, but something it forces us to do is take seriously the relation of Being to Set Theory and other foundations of Mathematics which is something we are not forced into. Mainly this is because the Ontologists don’t know enough math and mathematicians do not think about ontology. But Badiou prides himself on having gotten the mathematics down well enough to philosophize about it. What is wonderful about that is that Badiou uses Cohen’s work and generalizes it to give it an ontological bearing. And that helped me because at the time I was trying to figure out what the next level up of abstraction beyond General Schemas Theory and wondering if I needed to define that in order to ground General Schemas Theory. Basically using Cohen’s result in Set theory that says that there is some independence between certain axioms and the others, we can generalize this method and see that if something does not matter at the level of abstraction you are at if you changed it, i.e. if you cannot tell the difference if it changes, then it is not a transcendental constraint that you have to worry about at that level, and that frees us from being concerned with upper level transcendentals if they have no lower level effects if they are changed. This is an excellent point. So if changing the worldview structure did not change anything at the General Schemas Theory level then I did not have to worry about grounding General Schemas Theory in a higher theory of the structure of the worldview. And I decided that was the case so it freed me of a lot of work that would have been probably unnecessary.
But what you see in Badiou is the focus on Set theory as the basis for mathematics when there are actually several contenders like Category Theory (my personal favorite) and Mereotopology, etc. Seems to me we have to consider all the various possible foundations of mathematics and their intrinsic variety and then generalize from that. Badiou wrote a companion volume to Being and Event which I have not read yet but it applies Logic to Worlds. This seems implausible but we will have to wait to read it to criticize it.
But my view is this which I have formulated on my website at http://holonomic.net. Logos and Physus are the most important duals in the Western worldview, and Nomos is the non-dual interface between them. Logic is the Core of Language (the Physus of the Logos) and thus we can see that schematization is the core of the Physus (the Logos of the Physus as what Kant calls an A priori synthetic projection of order as the singular of spacetime). Thus Logic appertains to Language (the Logos) and its opposite is schematization which structures both language and also our apprehension of the intelligibility of organized things in spacetime. But the nomos is separate and that is where the math is. And by the way it is only Indo-European languages that have Being, so it is a special anomalous case. So to identify it with set theory is obviously wrong. It seems to me that mathematics comes first because Nomos is the nondual that is there before Logos and Physus come into existence and separate from each other as a duality. And we can represent logics as Topoi (the category for logics) and that means that we can have a purely mathematical view of the structure of logic prior to the existence of Logic as the core of Language.
However, once Logic itself exists then it plays over the various possible foundations of mathematics that make up our meta-mathematics. Meta-mathematics comes back from Logic toward Math and tries to understand its foundations, and so both Logic and Math each help us to understand the other. Both make sense of each other by providing a different context in which the other can be explored.
However, it is not that Logic and Phusis come from Mathematics as much as they are pre-constrained by mathematics. For instance, in Science we look for mathematical foundations of our theories and then look for anomalies that disprove our theories that are structured based on the mathematics. Theories that withstand this kind of disproof tend to have a very close operationalization through their connection with the math. Similarly, there are myriad logics, and understanding those logics and their differences help us to find different ways to interpret the math and to see the possible foundation for the math, and it helps us structure our theories better so that they are logically consistent, complete, clear, verifiable, validatable, and coherent.
Hope this helps to clarify the mutual elucidation of Math and Logic which happens in the context of philosophy of Science and in the pursuit of science. This is really at the core of our worldview. And it is something we really need to understand, and don’t completely understand as yet.