What purpose do mathematics and philosophy serve epistemologically (compared to sciences)?

Question

Kant did not consider them sciences, but meta-disciplines that study a priori conditions of doing science. Indeed, both mathematics and philosophy permeate all empirical sciences to varying degrees, in fact to complementary degrees it seems. The more there is of mathematics in a field the less there is of philosophy (sadly).

Much is made about the difference with empirical sciences. However, if one believes in the Platonic realm and some means of accessing it directly (like Gödel did) then both mathematics and philosophy become "empirical" sciences of that. Husserl gave an account of ideal intuition that fulfills both conditions and is free of fantastic elements in Plato. So this distinction seems to rest on philosophical preferences.

One could also argue that proof standards in mathematics differ only in quantity rather than quality from soundness standards for theoretical arguments in empirical sciences. Another issue often brought up is experimental falsification. But from Kuhn we know that theories are not exactly falsified by individual experiments, they fall if they broadly fail. That isn't that different from mathematics and philosophy, although mathematical theories or philosophical systems are not "verified" or "falsified" fruitful ones are perpetuated (like Lebesgue measure theory or Kant's critical method), and unfruitful ones are abandoned and become historical curiosities (like classical invariant theory or Berkeley's solipsism). For mathematics the opposite sides are argued here and in the top answer to Is Mathematics considered a science? Argument about philosophy is mentioned here.

Not surprisingly, Husserl wanted philosophy to be a science, even a "rigorous" science. Ironically, it is analytic philosophy that comes closest to fulfilling his wish. However, even he distinguishes formal and empirical sciences (corresponding to his formal and regional ontologies), and places mathematics and philosophy with the former. On the other hand, most existensialists would have none of that. Even grouping mathematics with philosophy is controversial, in academia mathematics is usually put together with sciences, and philosophy with humanities. However, even some existentialists would probably claim that they study something more fundamental than "human culture" (Heidegger comes to mind).

Is there something to the question that does not reduce to an argument about words? Accepting Husserl's formal/empirical terminology can the underlying common of "science" be identified without including "pseudo-sciences" (or should some of them be included)? And if mathematics and philosophy are (are not) sciences should they be (not be)? In case of the negative answer, how is the role of mathematics and philosophy similar/different to that of "sciences"?

Answer 1

[To avoid @Conifold's worry this will just be about words let me warn you ahead of time -- I accept 'science' as a term that applies equally well to all the things Newton considered 'science', including not just math and a big part of philosophy, but his extensive investment in Alchemy. If you cannot agree, this argument will remain unconvincing to you. I only choose Newton because it is English that has the strongest bias about the definition, and he is the epitome of early English science.]

I think that Lakatos in "Proofs and Refutations" makes a good case that however the field looks at itself, mathematics is verified for the rest of us by its applicability, and not by its internal consistency. (Without really stating it, since he is trying to stay in a dialog format. He makes the super-modernist ('Theta'?) look vapid and smug. He 'wins', but he loses our concern.)

Taken in that vein, it turns back into a science, but it is a science like linguistics or (non-Behaviorist) psychology, about internal human processes whose external effects are only source material and not subject matter. So I would not make the distinction between formal and empirical sciences so much as between exterior and interior focus. The cheesy way of putting this is "Mathematics is just the oldest branch of psychology."

One might claim that modern linguistics is not a science, and psychology must restrict itself to Behaviorism and neurology in order to be considered scientific. Even more would claim fuzzier sciences like anthropology or sociology either simply not exist as sciences, or they are way outside their own proper boundaries.

But all of those disciplines regularly make predictions about external facts. Math predicts that if you do your statistics right, you will not see huge contradictions between our predictions in practice. Anthropology predicts we will find certain aspects of certain cultures at dig sites eventually, that we may not yet have seen to date. If they turned out to be wrong most of the time, things would change, if only slowly. So they are doing the job of a science, even by a fairly prescriptive definition of scientific. Still, they are not particularly invested in those empirical facts, at root. They rely more upon complex, internal models of human experience or interaction, that have a different kind of coherence. And they therefore mainly see those facts as mere checkpoints, rather than focal material.

I would throw into this pot of 'internal' sciences even parts of aesthetics like music theory, some literary criticism, or comparative mythology, to the degree they theorize clearly about what makes things appealing in a predictive way (e.g. The Lydian Chromatic Concept of Tonal Organization or studies of recurring archetypes) and do not simply observe trends or invent labels, without attempting to explain the effectiveness of the art.

In a different way, across disciplines instead of between them, I would claim that Kuhn's notion of science should be seen as a complete continuum, and not a set of levels or defined periods, and that all of it is, to different degrees 'science'. Pre-science is science, and even a well-intended pseudo-science is science, just badly done. Philosophy, then, from that extended Kuhnian point of view, is the part of science that has not yet attained, or that cannot for deeper reasons attain, an initial stable paradigm of adequate coverage. It is therefore not 'a science', but it remains 'science' in a different sense. It is kind of 'science outside any given science' the same way there is a lot of water outside any given body of water.

In that sense, all the sciences are restricted parts of philosophy with accepted bases, and 'philosophy proper' is what is left that is more basic than the existing bases. Husserl might get his wish, but the part of philosophy that captured his goals would just spin off into a separate science the way Utilitarianism has become economics.

To me, the distinction of paradigmatic vs sub-paradigmatic addresses the apparent complementarity you note. A paradigm enables the application of other sciences to a field. Without one, there is less ability to get leverage from outside, because the points where leverage would be applied are moving. So the less strong the paradigm, the more often the science is thrown back into more basic questions, and the less likely any outside discipline is to bear on progress.

Answer 2

The three fields are so interdependent it's hard to separate them. I would generalize their interconnectivity like so:

Mathematics tests proofs to turn data into information; science tests theories to turn information into knowledge; philosophy tests positions to turn knowledge into wisdom.

It's one thing to know or understand; it's quite another to feel or to reason. This is why I consider philosophy one of the humanities (along with the arts, literature & religion et al.) more so than as a science. There really isn't much room for the human heart and individual expression in mathematics or science. And the human heart is where philosophy becomes transcendent...

Answer 3

The methods by which mathematics and science operate are fundamentally different.

Mathematics begins from axiomatic foundations, and extends from there by induction from axioms to arrive at results as true as the original axioms, which are, within a given mathematical system, true by definition.

Science attempts to map objective reality. It is prone always to having its theories overthrown by some new discovery. Induction remains useful as a guide, but it cannot serve as proof. Scientific proof is of a weaker sort: something has been observed to happen, and thus it is proven (subject to experimental error) that it happens sometimes.

Further to this, we do not doubt mathematics when our empirical data casts doubt on something that has always been considered mathematically true; we would rather assume our reality takes the strangest forms than doubt it. Bell's theorem leaves physicists choosing to abandon locality, realism or freedom, but at no point has someone suggested that sometimes numbers just add up differently than they usually do.

Proof differs therefore not merely in quantity and quality, but in kind. The difference lies in the validity of induction. Of course, how the human performing mathematics or science arrives at conjectures that require proof is a process that has a great deal of commonality across both disciplines. How they proceed from there is where the difference lies.

I have not mentioned philosophy, for, depending on the philosopher, it may operate under less rigorous rules than either of the others, and it is therefore more hard to generalise.