# Is Mathematics the Ultimate Culmination of Analytic Philosophy?

As a novice amateur, the similarities between mathematics and analytic philosophy seem striking to me. At least in a caricature view of analytic philosophy, it is the project of establishing the necessary and sufficient conditions for the validity of different claims given a framework of well-defined concepts. And, on the other hand, mathematics is also about exploring the implications of (i.e., exploring as to which claims are true or false given) a set of axioms and definitions. The similarity between the two projects seems difficult to ignore. I realize that one can make a distinction in that the well-defined concepts of analytic philosophy usually correspond to (or come out of) ideas arising out of the natural experience whereas the mathematical objects can be purely abstract. However, given the potentially infinite range of natural phenomena, in principle, one can imagine a variety of exotic concepts which can be subjected to the usual process of conceptual analysis/analytic philosophy. In this vein, it appears that mathematics is nothing but precisely this project--it is the conceptual analysis of all logically conceivable frameworks of concepts, the ultimate culmination of analytic philosophy.

I would like to know if this is a valid argument or a result of my misunderstanding of what analytic philosophy is. I would also appreciate if one can direct me to references relevant to the question of the relation between mathematics and analytic philosophy.

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Another way of stating the claimed similarity is that both mathematics and analytic philosophy seem to be in the business of exploring a priori truths given a framework of concepts or definitions and axioms. They do not, in and themselves, make statements whose truth might be subject to empirical evidence. However, when concepts from philosophy (or mathematical objects) are identified with aspects of natural phenomena (as an axiomatic step in science), if the identification is empirically correct, the necessary truths of philosophy (or mathematics) automatically translate into empirical truths that one can verify.

• I do not think that analytic phil is "about exploring the implications of a set of axioms and definitions." It is more about the elucidation of concepts. Jul 28 '19 at 8:42
• Is a logician a mathematician or a philosopher? Ask a philosopher and they will tell you that logic is a branch of philosophy. Ask a mathematician and they will tell you that logic is a branch of mathematics. To a digital engineer the table of possible logic states for a circuit is a "truth table". I hope your question stirs some interesting answers. Feb 18 '20 at 12:49

I think your proposition is actually reversed. Rather than mathematics being the culmination of analytic philosophy, analytic philosophy was an effort to rebuild philosophy on the already-proven ground of mathematics and the natural sciences. People like Frege, Russell, and (early) Wittgenstein leveraged mathematical methods to try to build a more sophisticated form of propositional logic that would dovetail nicely with empirical methods. It was all part and parcel of their desire to cut off what they saw as 'specious' metaphysical argumentation.

Mathematics is not itself philosophy. It is too formal, having no intrinsic value or meaning: qualities that are essential to any philosophical project.

• That is precisely what I am arguing that just like mathematics, ultimately, analytic philosophy is a formal structure of well-defined objects and they gain meaning grounded in reality only when they get identified with elements of nature. Jul 30 '19 at 7:47

I share your view of mathematics and metaphysics. My go-to example would be Russell's paradox, a problem Russell refused to see as a problem in metaphysics for to do so would have threatened his world-view, but which his contemporary Spencer Brown solves as a problem in metaphysics and mathematics by reference to Taoism.

One can do higher-level mathematics while ignoring metaphysics but not the the reverse, and the foundations of mathematics might as well be metaphysics.

I see them as much same sort of activity. Both require concept-definition, symbolisation and analysis using logical rules.

The problem of 'axiomatising' the Universe is logically equivalent to doing this for numbers or sets. Mathematicians who take this approach would include George Spencer Brown and Hermann Weyl. On this view Russell's paradox would prove that dualism and monism can never be fundamental theories.

Analytic Philosophy can be seen as the larger scaffolding that justifies the project of mathematics, within a philosophical setting. However, mathematics can be argued to exist in and of itself, and can easily support itself with the feedback from a socio-tehnological / evolutionary feedback, without the need for philosophy. Animal behavior researchers try to study rudimentary capacity towards mathematics in simpler animals like bees for example, and such a very basic capacity for mathematics need not even come from conscious mental activity it could be argued. Take for example, a microorganism that is very sensitive towards the Ph or salinity of the medium it can tolerate.

In human societies, analytical philosophy is another intellectual activity that lives the life and times of any intellectual inquiry, with its successes and limitations. However, I would disagree with @PeterJ that metaphysics is predicated on mathematics. Questions around the unreasonable effectiveness of mathematics, or the many to one problem, deal with something more global than symbolic mathematics in my view.