Research: for non-philosophers


Why this note?


I'm often asked by non-philosophers what is to do research in philosophy, in particular in philosophy of science and mathematics. Here is the answer I should give. I usually don't give it, or give it only partially, because this is not an easy, small-talk type of question, contrary to what the questioners assume. I stress that this is my answer (as a philosopher belonging to the analytic tradition), not the answer. Other philosophers will probably give a different answer. (And this may already be the first thing to learn about philosophy, that philosophers rarely agree with each other.)


So, let's begin with a deep, eternal truth: when one encounters problems, one tries to solve them. Scientists try to solve scientific problems; that is, to answer certain empirical questions. (Physicists, for instance, ask questions like these: Why do unsupported bodies fall? Why does water freeze when cooled? Why does the sun shine?  Mathematicians have problems to solve as well: Is every even number greater than 2 the sum of two primes? And so on.) Philosophy, too, deals with problems, and tries to answer some questions. 


However, when it comes to characterizing these questions, things are rather complicated. This is so for three reasons, detailed below. These considerations also apply to the specific domains of philosophy of science and philosophy of mathematics, and I will get to them after these more general remarks.


The first reason is that it is not immediately clear what counts as a philosophical problem (or question; I use these two words as synonymous here). 


The very nature of such a problem is itself subject to philosophical debate, and this is one thing that makes philosophy unlike the sciences and mathematics. However, as we'll see, these three domains are otherwise intimately connected. Some of the deepest philosophical problems appear within science and mathematics (or regard their applications). There are issues, especially concerning the foundations of a scientific field, when any attempt to separate the scientific-technical aspects from the philosophical assumptions appears as arbitrary. When it comes to moral and political issues –  abortion, capital punishment, distribution of scarce resources among people, etc. –  this entanglement is even more radical. 

(Part of my interest in the 20th century philosopher Ludwig Wittgenstein is motivated by his attempts to rethink the very idea of a philosophical problem.)


So, it is important to notice this dissimilarity: unlike philosophers, scientists and mathematicians work with (comparatively) well-posed problems and questions. But note that saying this is not fair to philosophy. Because it takes work and reflection of a philosophical nature to bring a question-problem to a form that allows the scientists to tackle it by non-philosophical / scientific means. That is, without questioning the questions first, i.e., without clarifying them – an activity of a philosophical nature, often not recognized as such! – science and mathematics would not be the amazingly successful enterprises they are today. (How this questioning and clarification is done, more concretely, is, I'm afraid, not possible to detail here.)


The second source of difficulties is that even if we manage to formulate a philosophical problem, it is far from obvious what it means to solve it. Many philosophers will acknowledge that no major philosophical problem is currently solved. Or, more depressingly, that none will ever be solved! Worse yet, there is no consensus as to whether any progress has been made on any major philosophical problem. But this may be too quick, since what constitutes progress in philosophy is a philosophical question itself (unsurprisingly). For one thing, progress in philosophy does not have to be like progress in science and mathematics. (Is there progress in music?) This is thus another feature of philosophical problems that makes them unlike the problems in empirical science and mathematics. In these fields, we can tell right away whether a problem has been solved, whether a question has received a true answer. Or so it seems; as you may already suspect, there are major philosophical complications with the notion of truth, both in science and mathematics. I'll leave them aside here.


(Back to Wittgenstein for a moment, an idea he pursued was that while philosophical problems can't be solved, it may still be possible to avoid them. Yet, again, how to avoid a philosophical problem is itself a philosophical problem – and realizing this may appease his critics, who complained that adopting this 'therapeutic' strategy means that we should abandon doing philosophy: it does not.)


Finally, the third reason: philosophical problems are special because even if a philosophical problem is formulated and a convincing solution to it is proposed, it is still not at all evident what we gain from this whole exercise. One may even wonder why on earth should we consider such problems to begin with?! This is so since while they are connected to the flux of practical life (including the scientific life), their depth and generality also separate them from it. 


The very tendency to ponder these issues is, I reckon, a bit of a mystery. It could be a natural, universal human inclination, as natural as other human needs (e.g., to make friends). Yet only a small minority of people display it. In the end, it may be that the (only?) gain of engaging in the philosophical exercise is nurturing one's ability to think clearly – and this means: to see distinctions and connections – in any subject matter. Thinking clearly is not as easy as it seems, while few things are more important for a human being, especially a young one. It is not stressed enough how much society as a whole has to lose when we, and our leaders, lack this capacity.


For these three reasons, it is perhaps more accurate to say not that philosophers solve problems, but rather that they follow their inclination to study, to understand various philosophical puzzles. And part of this work is done for clarity's own sake. (To bring Wittgenstein in again, one should ask if there is anything wrong with seeking clarity for clarity's sake? That is, with clarity being not just a means to an end, but the end in itself? Another philosophical question!)


Two philosophical problems


Enough preparation. Here are two examples of philosophical problems. Look at them in light of the difficulties signaled above.

(Philosophy is sometimes described as the art of drawing distinctions – as a condition to think clearly.) Note that now we have an answer to the question asking what is to do philosophical research in this area: it is to try to address this question (among other things).

Well, a good start would be to recognize that our actions and decisions are the result of the workings of our brains. Our brains are physical things, since they are made of atoms and molecules. And the behavior of physical things is entirely determined by the laws of physics (and chemistry, biology, etc.) Hence the workings of our brains, and thus our decisions, are fully  determined by these laws, just like the behavior of other physical objects and processes (i.e., think of what happens when you play billiard or pour milk in your coffee). If so, the philosopher's challenge is that nobody is (and has ever been) responsible for anything – that is, blameworthy or praiseworthy of anything. One has to admit that this is a worrisome conclusion!


Again, just as above, to do philosophical research on this issue consists, among other things, in trying to meet this challenge. And in this case it is easier to see why some philosophers may seek to collaborate with neuroscientists, or other experts in the working of the brain. Note, however, that other philosophers argue that tackling the problem of free will, as they understand it, does not require such expertise at all.


These are only two examples of typical philosophical problems. There are of course others – about what is right and wrong, about the best organization of society, about aesthetic value, about science, mathematics, and so on. But, recall, it is not a trivial matter to accept that a problem is philosophical. To illustrate this, here are some questions I'm often asked when people find out that I do philosophy: What is the meaning of life? Is there an afterlife? Is there a God? etc. 


It's hard to describe the disappointment of the questioners when I confess that I'm not sure that these are philosophical questions, maybe not even questions at all (despite their grammatical form). They may be turned into such questions; but, as usually posed in casual conversation, I don't think they actually ask anything. However, they do have a function: to convey existential angst, and this is not something to belittle.


So, I will set these questions aside and confine my remarks here to the problems and questions which have a clearer philosophical status (like the two above, about knowledge and free will). These questions, and others like them, are handed down to us from times immemorial. Many cultures around the world faced them once they reached a certain level of intellectual development (perhaps not in the specific formulations given here). The first problem belongs to what is technically called 'epistemology'; the second to 'metaphysics'. They are relevant for human endeavors, and will be with us as long as the human form of life will exist. For instance, if you are ill, you want your doctor to know the cure, not only to believe she does. Or, if someone stole your possessions you would be puzzled, even enraged, if the judge let the thief go, on the grounds that they lacked free will (e.g., their misdemeanor was a result of their upbringing, of their social circumstances, etc.) 


We can now return to answer our main question: what is to do research in the philosophy of science and mathematics? 


In my view, the point of this kind of inquiry has been summarized quite well in the Toulmin quote I reproduced on this page: to offer outsight. That is, to address a family of questions which strictly speaking fall outside scientific or mathematical expertise, and yet are often perceived as being in need of an answer. Given the nature of these questions, the scientists themselves are not in the position to tackle them qua scientists. Thus, to address them properly one needs a different set of conceptual tools and a different approach. This is exactly what philosophy of science provides - and, of course, not only philosophers of science can employ these tools, but they are available to everyone, scientists included. Thus, the primary goal of philosophy of science is not to produce (more and better) science in a direct way, i.e., to offer scientific insight, but to contribute to a better understanding of the assumptions, methods and results of science, i.e., to provide outsight.

(As always, be warned that this distinction is only a first approximation; it can, and should, be itself brought into question.)


Here are two concrete issues discussed in philosophy of science.


Consider a scientific claim that has become commonsense today: a drop of water is made of many, much smaller identical components (water molecules), having a certain internal structure. This structure – two atoms of hydrogen and one of oxygen – is itself described in terms of chemical bonds. And these, in turn, are described in quantum mechanical terms. A question that the philosopher (of science) will ask here is: Do we, and the scientists, really know all these things?

 

Sure enough, following the scientists, we believe them. But, since there seem to be a difference (recall: what is it?), the question still stands. (Or, to be even more careful, we accept them.) One may already notice that the philosopher's question above is somewhat odd: isn't this a question that scientists themselves have already asked when they made these claims? And, haven't they already answered it satisfactorily, based on many years of doing careful experiments and tests? ('Yes, we do know these things.'). Here a philosopher will tell you that this can't be the whole story, for (at least) two reasons.


Before we get to these reasons, note that this inquisitive attitude – the refusal to take anything for granted, the impulse to look underneath the surface – is a distinctive feature of a philosophical mind. Possessing it doesn't presuppose an institutional affiliation to a philosophy department, or having earned a philosophy degree (although this kind of training may help). But also remember that this attitude may be dangerous - Socrates was murdered because of it! -  and sometimes even counterproductive. So, I'm of the view that it is perfectly okay not to have it all the time. Of course, this inclination is a sign of intellectual maturity and depth; other things being equal, people who display it wisely rank highest in my eyes. What I'm saying is that not everyone should think like a philosopher, and definitely not all the time. On the other hand, it is important to recall that among scientists and mathematicians some of the greatest exhibited philosophical acumen; that is, their scientific/mathematical contributions contain a significant philosophical-conceptual element. (The entire list would be quite long, so here I'll only mention, in chronological order, Galileo, Newton, Leibniz, Cantor, Poincaré, Bohr, Einstein, Noether, Wigner and J. S. Bell.)


Let us get back to the first reason to be philosophically anxious in relation to science. It has to do with what many people – scientists included – take (simplistically, the philosopher will add) to be the method of science: the experimental testing of theories. The idea is well-known, and goes like that. One proposes a theory, or explanation, of a certain phenomenon: 'this happens because of that'. Then one tries to show that their theory holds, or that it is not just their own subjective belief, but it constitutes objective knowledge. One sets out to make observations and experiments to test the theory. Here is where the philosopher intrudes, and asks:


The worry here is that the tests made today may not have the same outcome tomorrow, and in the future. Thus, a hidden – albeit in plain sight! – assumption laying at the foundation of the entire empirical science has been revealed, namely that Nature must operate in a uniform fashion. This means that what we have not observed is like what we have observed, or that the future will be like the past. It is the identification of the difficulties around this assumption that counts as a major philosophical achievement. (It is attributed to the 18th century philosopher David Hume). The problem it generates – are we justified to believe that Nature is uniform? – goes by the name of 'the problem of induction'. (Curiously, Hume himself never used this word.)


Another reason of philosophical concern is related to the first, and comes from considering the history of science – a subject very seldom part of scientific curricula in universities today. Hume's question may be very difficult to answer, but inductive reasoning generally works. And the past historical record seems to indicate that many (if not all) of the things that the scientists of the previous eras have said turned out to be, by the current scientific standards, false. (Maybe this is why this pessimism-prone subject is neglected in the education of the new generations of scientists?) So, we suddenly face a problem: if (great) scientists of the past made false claims, is what the scientists say today true? The philosophical term of art describing this puzzle is somewhat intimidating, but quite apt: 'the pessimistic meta-induction'. (It should be clear why this is some kind of induction, and why it is pessimistic. The issue was discussed at length by Poincaré). 


Thus, some of the difficulties of a philosophical nature are these: 


These are just two, and rather general, issues that philosophers of science ponder on. They also work on other questions like these, some going quite deeply into various scientific specializations. There is philosophy of physics, of biology, of neuroscience, etc. These, in turn, branch off into even more specialized fields. For example, within philosophy of physics, there is philosophy of quantum mechanics, of statistical mechanics, of space-time, of condensed matter, etc. As is clear, some of the questions above touch upon the relation between science and society, and some philosophers tackle this as well – often in cooperation with other experts, e.g., economists, healthcare professionals, sociologists or anthropologists. I should also add that I'm under the impression that, alas, the philosophical work in these more technical fields of philosophy of science is, with very few exceptions, virtually unknown to science students and professional scientists. This is a pithy since – and this may sound presumptuous coming from a philosopher – philosophers working in these fields often display an understanding of the scientific material from which even the best practitioners could benefit: outsight can lead to insight.


Before we move on to mathematics, let me say this: 


Now, what are some research questions in the philosophy of mathematics?


We become familiar with (what I think is) the essence of mathematics – formulating proofs – very early on. Right after we learn how to count, we calculate sums and multiplications by applying basic rules; then we get to prove some simple results, e.g., that the product of two odd numbers is always an odd number, or even more significant theorems, that there are infinitely many prime numbers. Yet, despite this thorough and early familiarity with mathematical matters, at some point one may stop and ask:


Again, it is okay if this question never gripped you. No worries: you can be a good mathematician (or accountant, or dentist, or plumber, or whatever) without ever thinking about it. As far as I can tell there is no reason why one should consider such a question. (I suspect that this holds more generally: no question – be it scientific, mathematical, philosophical, etc. – is so urgent that one should, or must consider it. But, is this so? Another topic for philosophical reflection!)


Yet, as it happens, the question does bother some people. (Why? again, I have no answer to  this question!) And, as expected, there have been attempts to address it. A first step is to refine it, and ask what kind of things numbers can be such that we get to prove (i.e., to know, not only believe!) all sorts of amazing things about them – e.g., that there is an infinity of them. Importantly, this is knowledge that, unlike our knowledge of empirical matters, is certain. (Euclid, the ancient mathematician, established results that, unlike Newton's, are not rejected today.)


The question above is perhaps the fundamental issue in the philosophy of mathematics. It is not easy to answer it. Just try. Great minds, such as the philosopher-logician and mathematician Gottlob Frege, struggled with it at the end of the 19th century, as well as several other unbelievably smart mathematicians-logicians-philosophers, people of the stature of Georg Cantor, Richard Dedekind, Bertrand Russell, David Hilbert or John von Neumann. Here's an opinionated summary of some of their answers, meant for a general audience.


Now the floodgates of philosophical curiosity about mathematics are wide open! A few more such questions, from a longer list:

In the unlikely event that you have made it this far, your reward is to hear about two more research questions in the philosophy of mathematics:


Final thoughts


If you are interested to read contemporary professional philosophy – a daunting task unless you have studied the subject at upper-undergraduate level! – then here is a list of ranked 'generalist' journals in the anglophone analytic world (as with any ranking, take it with a grain of salt). Occasionally, they publish work in philosophy of science and philosophy of mathematics. But most of this work appears in 'specialist' journals: Philosophy of Science, British Journal for the Philosophy of Science, Philosophia Mathematica, European Journal for Philosophy of Science, Studies in History and Philosophy of Science, Biology and Philosophy, Foundations of Physics, Perspectives on Science, International Studies in Philosophy of Science, Foundations of Science. In case you suspect that you have some philosophical insight to communicate to the world, test your idea by submitting it, in the form of a paper, to these journals. Keep in mind that the standards of acceptance are very high. Your paper will be sent to two anonymous expert referees (if it survives the appraisal of the handling editor). They will report back to the editors on whether your contribution is likely to be of any interest to the readers of the journal; in some cases, you will be sent their reports (after at least one month's wait, typically more). For the top venues, the acceptance rates are minuscule, somewhere between 2-10%, so don't hold your breath.