What is the difference between math, science and philsophy?

I’ve been listening to the Philosophy Bites podcast recently. One from a few years ago consisted of answers from philosopher’s to the question posed on the spot and without time for deep reflection: What is Philosophy? Some managed to give precise answers, but many struggled. I think one source of conflict they faced as they answered was that they didn’t know how to separate the question of what philosophers actually do from they should be doing. However, I think that a clear distinction between science, math and philosophy as methodologies can be specified precisely. I also think that this is important because practitioner’s in each subject should be aware of what methodology they are actually using and what is appropriate for whatever problem they are working on.

Here are my definitions: Math explores the consequences of rules or assumptions, science is the empirical study of measurable things, and philosophy examines things that cannot be resolved by mathematics or empiricism. With these definitions, practitioner’s of any discipline may use either math, science, or philosophy to help answer whatever question they may be addressing. Scientists need mathematics to work out the consequences of their assumptions and philosophy to help delineate phenomena. Mathematicians need science and philosophy to provide assumptions or rules to analyze. Philosophers need mathematics to sort out arguments and science to test hypotheses experimentally.

Those skeptical of philosophy may suggest that anything that cannot be addressed by math or science has no practical value. However, with these definitions, even the most hardened mathematician or scientist may be practicing philosophy without even knowing it. Atheists like Richard Dawkins should realize that part of their position is based on philosophy and not science. The only truly logical position to take with respect to God is agnosticism. It may be probable that there is not a God that intervenes directly in our lives and that probability may be high but it is not a provable fact. To be an atheist is to put some cutoff on the posterior probability for the existence of God and that cutoff is based on philosophy not science.

While most scientists and mathematicians are cognizant that moral issues may be pertinent to their work (e.g. animal experimentation), they may be less cognizant of what I believe is an equally important philosophical issue , which is the ontological question. Ontology is a philosophical term for the study of what exists. To many pragmatically minded people, this may sound like an ethereal topic (or worse adjective) that has no place in the hard sciences. However, as I pointed out in an earlier post, we can put labels on at most a countably infinite number of things out of an uncountable number of possibilities and for most purposes, our ontological list of things is finite. We thus have to choose and although some of these choices are guided by how we as human agents interact with the world, others will be arbitrary. Determining ontology will involve aspects of philosophy, science and math.

Mathematicians face the ontological problem daily when they decide on what areas to work in and what theorems to prove. The possibilities in mathematics are infinite so it is almost certain that if we were to rerun history some if not many fields would not be reinvented. While scientists may have fewer degrees of freedom to choose from they are also making choices and these choices tend to be confined by history. The ontological problem shows up anytime we try to define a phenomenon. The classification of cognitive disorders is a pure exercise in ontology. Authors of the DSM IV have attempted to be as empirical and objective as possible but there is still plenty of philosophy in their designations of psychiatric conditions. While most string theorists accept that their discipline is mostly mathematical, they should also realize that it is very philosophical. A theory of everything includes the ontology by definition.

Subjects traditionally within the realm of philosophy also have mathematical and scientific aspects. Our morals and values have certainly been shaped by evolution and biological constraints. We should completely rethink our legal philosophy based on what we now know about neuroscience (e.g. see here). The same goes for any discussion of consciousness, the mind-body problem, and free will. To me the real problem with free will isn’t whether or not it exists but rather who or what exactly is exercising that free will and this can be looked at empirically.

So next time when you sit down to solve a problem, think about whether it is one of mathematics, science or philosophy.

8 thoughts on “What is the difference between math, science and philsophy?

  1. The little philosophy group i go to discussed last week whether brain science made philosophy obsolete. I dont think there was a consensus; one person took the ‘Dennet’/mechanistic view—showing me a card which had brain states on one side with arrows pointing to mental states on the other (and no arrows the other way—eg ‘top down causation’ (p davies—who i think got a big NIH grant, or G Ellis). This person is also a (liberal) Christian, and i asked how that came from his theory or diagram. He said basically because its true (true brain states are christian brain states). Sam Harris would argue the true ones are atheistic ones.
    I also wondered which brain states were involved with finding or making that card; i think he needed a bigger card (or big data). .

    I tend to think of philo, math and science as related the way ice, water and steam are related, or morphogenetic states or animals—they are connected by a symmetry breaking bifurcation.
    (I once put this in a paper —called ‘forces, forcing [from set theory] and information’ which was not very rigorous and also ignored and ridiculed—though sometimes that attitude is due to people worrying about things like ‘if god is dead or nonexistant , then i won’t have a job or any respect’). (I notice someone at CMU in philo got a huge grant from the military to look at ‘homotopy type theory’, which connects language, ontology and topology in a slightly different way (though the ontology likely is already there, in the way electrons may have consciousness–the moral of the story is its funding source— ‘what goes in, comes out’ (curie prigogine).

    I follow Tipler’s classification in ‘the physics of immortality’ and classify myself as an epistemological reductionist, so i’m effectively agnostic (the plausibly deniable position) on whether there is an ontology, or even empirical world. Its easier just to ‘set all these parameters or constants equal to 1 or 0’).)

    From this view in a sense they are all obsolete except as conveniant conventions, in the way one might say today i’m doing chemistry, tomorrow biology and the next day physics even though they are all the same thing with different notations (as are say discrete versus continuous math, or various logical types (russel)).
    Of course, these may not ‘commute’ so one may have ‘new constants of the motion’ (suggested by robert rosen of the rashevsky school of theoretical biology, and implied by people like P W Anderson—‘more is different’— or R Laughlin (see eg ‘the crime of reason’). The issue is basically whether math is physics or the reverse. (Excercize left for the reader).

    Regarding the issue of ‘free will’, PNAS in 2010 has a discussion of this (and crime also) due to Cashmore of U Pa.
    It does appear there may be a little bit of inconsistancy (which one would expect for any open system or symbolic one leading to ‘skolem’s paradox ) since the issue is rephrased as ‘who or what is excercizing that free will’. (I would add ‘doing exactly what’. )

    I thought the FCC decided this yesterday—nothing is free (there’s no net neutrality, neutral evolution, etc.), not your axioms or your choices. One could ask Cashmore of U Pa, but to do that might require more cash. .



  2. p.s. the discussion on november 2013 FOM-list on ‘wigner’s unreasionable effectiveness’ idea is another frequency in the spectrum of views.


  3. Hum, no, I have a different argument “against God”: It is silly an meaningless to hypothesize something like God because this does not explain anything, replacing a mystery by another, who created God?
    Turtles all the way down…
    Pure paranoia!


  4. It could be said that the difference between U and I is one of paranoia and mathematics….Do not be hasty to draw instant conclusions to one or the other in their simplest, instant impulses of thought that immediately enter your mind..But think / feel about the effect of what one has on the other…and how each has an effect on your life…You will then see the difference between U and I….Someone who feels they know…


  5. Carson, someone not familiar with your research or this blog actually recently shared this post with me, which I found to be a neat coincidence.

    In additional to the math-philosophy-science triangle, would you consider engineering to represent a separate precise methodology such that there is a math-philosophy-science-engineering square? Engineering also utilizes math, science, and philosophy, but separate from them has an overarching goal of building tools, solutions, and algorithms for specific problems.


  6. @wally I actually had to reread what I wrote. I’m not sure I would call it a square as that would make other subjects feel left out but I think engineering applies knowledge to create new entities and thereby add to the ontological list. In that way, I think it is closer to mathematics than philosophy or physics. One of the shocking things I discovered in graduate school was that engineering and economics were more mathematical than physics, or at least they were more integrated into main line math. Physics kind of lives in its own little math world.


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