My question is whether a lack of knowledge about formal mathematics or theoretical science in general would have an impact on a philosopher's ability to think and make judgments.

Why should a philosopher acquire a deeper understanding of natural sciences or develop mathematical and scientific ways of thinking?

  • This is totally my opinion but I seriously think that a good philosopher must have a good grasp of math and physical world. Knowing math is like a swimmer cross training on weights to learn to isolate and train individual muscles. Math if nothing else trains and tunes the mind. As far as not knowing natural science how do you philosophize about human nature if you don’t know how our surroundings work. At the other hand you will find that all the big philosophers had a funny opinion on how some thins worked.
    – user267
    Commented Jun 20, 2011 at 10:27
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    I've never heard of anyone complain that they knew too much math or science, but I have seen many limited by not knowing enough. Commented Oct 22, 2012 at 4:45
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    The laws of logic and Maths are to a great deal intertwined.
    – Neil Meyer
    Commented Mar 19, 2013 at 11:44
  • A solid background in Cognitive Science isn't to sneer at, either.
    – shieldfoss
    Commented Jul 17, 2013 at 7:55
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    As someone who's dabbled in both science AND philosophy, I like to think of science as the framework, and philosophy as the thinking about the framework. You might not always need the science, but the more you know the better your philosophy will be.
    – Cdn_Dev
    Commented May 11, 2015 at 15:55

11 Answers 11


This depends very much on the area of philosophy. If you're interested in philosophy of quantum mechanics, for instance, you need at least undergraduate level training in physics (and the mathematics that entails). If you're doing ethics or political philosophy, then maybe the need for that sort of knowledge is lessened (although knowledge of some basic logic and some economics would have vastly improved several talks I've had to sit through...). Philosophers of mind need a good understanding of neuroscience and possibly some psychology.

Now these are just examples of the maths/science that philosophers need in order to usefully contribute to an area. There is the broader question of what understanding could be recommended even if it's not a prerequisite for doing the philosophical work. A basic knowledge of mathematics and science is always a good thing. Having studied maths gives you a particular way of thinking through problems that Intro to Logic just doesn't. That extra facility with thinking logically is always useful.

Then there's the even broader question of what any right thinking person should know. And as a consequence what every philosopher should know. This includes, I think, some basic physics: an understanding of electricity; mechanics and kinematics; conservation of energy... Some basic maths: what a function is; how to calculate a percentage; how to read statistics that crop up in newspapers and adverts (the difference between a relative and an absolute increase...)

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    then as a programmer, what areas of philosophy can i tackle ?
    – Belun
    Commented Oct 3, 2011 at 21:23
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    It's impressive how biology - the "central science" - is usually ignored. That's why so many philosophers are still discussing the biblical god - even 150+ years after Darwin...
    – Rodrigo
    Commented Jan 28, 2015 at 14:44
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    @Rodrigo, why is Biology the "central science"? what's so central about it? (i might expect to see Physics as more central.) Commented Jan 28, 2015 at 23:09
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    Biology is connected with our lives in much more senses than physics do. Morality may be explained by biology, not by physics. Emotions, education, may be observed in other mammals, not in protons. And so on.
    – Rodrigo
    Commented Jan 28, 2015 at 23:13
  • I'd add the life sciences to what a thinking person should know. Given that you yourself are alive, I don't know how this wouldn't be the most fundamental aspect of your pursuit of knowledge. I come from a science background, and physics as well as the life sciences make up the basis of pretty much everything else I understand.
    – Cdn_Dev
    Commented May 11, 2015 at 15:38

I would expect a philosopher to know about propositional and first order logic, and proof techniques. It would be worth your while to read Herbert B. Enderton's book A Mathematical Introduction to Logic. That may sound a bit harsh, however I encounter the use of them in my everyday life.

Countless times, I have seen people use phrases like "this is the proof of that" or "it is clear that every X stems from Y". However, when I look at what they actually are talking about I see neither a proof nor the clarity they are talking about even though the words like proof or every (∀) are well defined under mathematics and mathematical logic.

Also, when one listens to a person good at public speaking (like a preacher or a politician) one can easily think "Wow, he is damn sure right!" However, when you dissect each of his sentences and express it in first order logic (when you can), you can clearly see the fallacies, mistakes, and blabber.

  • True, however I would argue that first-order-logic is not the only means of proving the things that they prove.
    – Pacerier
    Commented Jul 10, 2014 at 7:42
  • I agree, that's why I didn't say that it's the only thing they should know. Commented Jul 11, 2014 at 6:08
  • I meant "propositional and first order logic" can be optional, because they are not the only means of accomplishing what they accomplish.
    – Pacerier
    Commented Jul 13, 2014 at 4:09

I respectfully disagree with Seamus, mostly because of his rather audacious use of the word "need". I don't think a philosopher needs to know anything in the sciences—in some cases they ought to in some regards, but they don't need to. The first (ancient) philosophers of course started in exactly this way, and they got along fine. Even centuries later, when the sciences were moving full steam ahead, philosophers like Immanuel Kant* and Hume and many others proposed many great theories without the aid of science that are still accepted today, because they were founded on logic and reason; phenomenological experience being the only thing they needed to start.

Furthermore, in some ways scientific theories can be perniciously misleading. Science is, after all, a collection of constantly revised theories that go through rather large changes from time to time, called paradigm shifts. Basing your arguments on potentially false theories in science could have obvious potential drawbacks. There were points in history with concepts like the flat Earth, the Sun revolving around the Earth, Atlas holding up the sky, souls being located in the pineal gland, etc... To have based a philosophical theory on such things would have been disastrous.

That said, I think it is the duty of every philosopher to never stop learning more, and those who continue to seek knowledge will always be more well-rounded and more prepared than those who do not. As Seamus and others have suggested, if you are interested in a particular field of philosophy that relates directly to a science (say, the philosophy of quantum mechanics), obviously it behooves you to learn about said field from as many perspectives as possible. Outside of that, the bad philosophers and the good philosophers will always be divided simply by their ability to form rational conclusions**.

*I'm aware the Kant was a scientist and taught many classes in the sciences. But he also created many theories regarding the scope of human knowledge and the limits of the human mind when there was no formal understanding of psychology in his time.

**Note that I don't think one's ability to use first-order and predicate logic makes a difference in their ability to reason; that is, the ability to form logical conclusions and to tell when an argument is invalid is independent of being able to use the formal notation.

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    This is a false view of science. Science is just thinking about nature done right, and if you don't know how to do it, you never learn how to think.
    – Ron Maimon
    Commented Apr 17, 2012 at 6:36
  • The problem is not everyone does it right, hence the revisions. We are, after all, only human. Furthermore, paradigm shifts can occur from properly done science; some discoveries radically change our previous understanding of the universe not because the old science was bad, but because we've just learned something new.
    – stoicfury
    Commented Apr 17, 2012 at 16:07
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    @stoicfury: The first (ancient) philosophers of course started in exactly this way, and they got along fine. I am not sure I understand what this way means. Do you mean that Ancient philosophers did not know anything in the sciences? I would have said the opposite, since Ancient philosophers were (also) the scientists of their time and the separation between philosophy and science only occurred (much) later. To take Ancient philosophers as examples that one can mind one's own business without knowing any science seems rather odd.
    – Did
    Commented Aug 30, 2012 at 11:45
  • @did - Yeah I realize now I worded it somewhat confusingly; I meant that the "first" philosophers obviously didn't have extensive backgrounds in math and science because the hard sciences (as we know them, scientific method and all) came much later before they were developed into formal systems. Sure, in a liberal sense they were scientists (I prefer to think so myself), but in the relevant sense they did not have the kind of background the OP is asking about. And I don't think that handicapped them at all, based on some great works I've read dating even before the Pre-Socratics... :)
    – stoicfury
    Commented Oct 22, 2012 at 2:34
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    I would add that additionally, the ancient philosophers got a lot of things wrong because they didn't know science. Mechanics got off the wrong foot for literally centuries because Aristotle thought that the circle was an elemental part of movement and directions. When Newton realized that things move in straight lines unless acted upon, mechanics underwent an amazing revolution.
    – shieldfoss
    Commented Jul 17, 2013 at 7:58

I absolutely vigorously advocate that all self-respecting philosophers know at least the basics of Grothendieck's categorical view-point of moduli stacks. It is both fundamental, and has implications about many important questions in philosophy. Some would argue with me, but I think it dwarfs Einstein's relativity as the most important philosophical advancement of the 20th century.

Grothendieck generalized geometry to the point of it coinciding with number theory (see here for example, but there are many others). The point of moduli stacks is the treatment of classification problems geometrically, in the proper generalization of geometry (think, as a baby example, an actual shape where each point represents an object you classify -- however the idea of stacks uses both a generalization of "classification" and of "geometry"). As a first step towards understanding this, I would look at, say, moduli of curves.

These generalizations completely transformed how people think about numbers, and how geometry is perceived (meaning that they are both projections of the same abstraction). Einstein's relativity, in comparison, is just the realization that spacetime is a 4 dimensional pseudo-Riemannian manifold with a specific metric - an enlightening idea to be sure, but still in a mathematical language that is standard to the way we think of geometry (there's nothing new about pseudo-Riemannian manifold, there's just something new about space being one).

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    You're trolling, right? It takes many years of training en route to becoming a professional mathematician to understand moduli stacks, leaving very little time to also become a professional philosopher.
    – Jeff
    Commented Jun 29, 2011 at 3:31
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    Can you give any ideas as to how 'moduli stacks' are important for a philosopher to know about, more important than say elementary logic, neuroscience, evolution, physics, etc. The things you said about moduli spaces have very little (OK, -no-) bearing directly or indirectly on philosophy. Can you explain in your answer how this very specific mathematical topic is all relevant to any area of epistemology, metaphysics, ethics, etc, -anything- outside of mathematics? Couldn't you get the same benefits from category theory, point-set topology, or really any focused area of math?
    – Mitch
    Commented Aug 16, 2011 at 18:10
  • I have to disagree, Mitch. Elementary logic is, well, elementary - so clearly everyone should know it. And the rest (evolution in particular) are impressive discoveries -- but they are discoveries that have empirical evidence. The theory of moduli stacks is a philosophical effort that is not only extremely impressive, but goes (successfully!) to the core of what a space is. It is evidenced in the proof of the Weil conjectures that it is indeed the natural notion of space, even though this notion has no points.
    – Rom
    Commented Sep 14, 2011 at 3:34
  • I see no other philosophical effort as being either as impressive, nor as successful. Can anyone honestly dispute that? (As for your last comment: category theory was merely a notation before Grothendieck's invention of abelian categories, and point-set topology is merely the usual sense of space. I think you seriously misjudge the importance of the discovering of moduli stacks)
    – Rom
    Commented Sep 14, 2011 at 3:36
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    That is the innovation, and it is extremely philosophically significant. The idea of space that appears to be most natural doesn't even contain points! It unites our counting (integers) with out perception of geometry (varieties).
    – Rom
    Commented Oct 21, 2011 at 15:17

Unless one is intending to actually do research in the sciences, I would hesitate to say that a philosopher needs a great deal of formal mathematics. Mathematicians themselves do very little formal mathematics (despite appearances); most of the work is in actually understanding the concepts & how they relate amongst each other, and their relative importance, which questions are important and why. Formal mathematics is a tool which helps & hinders this - rather than the thing itself.

After all, the philosophers first came up with the idea of atoms (which has been applied to matter, then to energy - the 'quanta' and perhaps in future to space-time).

Another example would be the Liar's Paradox that seeded Gödel's discovery of the his incompleteness theorems. He was astute enough to understand its ramifications and turn into formal mathematics.

  • It was democritus who came up with atoms, and it is not clear that democritus would be classified as a phiosoper or a scientist. Aristotle is a philosopher for sure, he is the philosopher's philosopher, and he killed the idea of atoms for two thousand years, using politics. This is how philosophy has worked in the past--- politics suppressing truth. The ideas that survive in philosophy are those that are promoted by the ruling class to serve their interests, not those which are best reasoned. This is why mathematics is essential-- can't fool math.
    – Ron Maimon
    Commented Apr 17, 2012 at 6:39
  • @Ron: How would Democritus experimentally verified his theory with the technology of his time? Politics does both - it suppresses, and it opens up. Commented Apr 17, 2012 at 6:52
  • This is what is remarkable about Democritus--- his arguments are all recognized today as correct compelling evidence for atomism. The first was the existence of sharp phase transitions, like ice melting. A continuous conception of matter leads one to believe that ice should gradually soften. Democritus believed atoms were in motion, and that melting was dissociation of an ordered crystal pattern. It is statistically true that when there is a large number of atoms, statistical phase transitions become sharp. This is an amazing insight for 300BC, it was not fully appreciated
    – Ron Maimon
    Commented Apr 17, 2012 at 7:03
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    in a small enough collection of passionate honest people, political persuasion tools are irrelevant. You only need sound arguments. That doesn't translate to large communities, where you need to persuade large masses of people who only have a passing interest, and cannot be bothered with the details.
    – Ron Maimon
    Commented Apr 17, 2012 at 7:41
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    @MozibarUllah: You can understand the sharp transitions as follows: consider an very large number of people and a few of them have a disease transmitted by handshaking that is infectious for 1 day. If the average number of handshakes in a day is less than 1, the disease will die out. If the average number of handshakes is greater than 1, everyone will get the disease eventually. This is a 0-1 property of large numbers, and it leads to sharp transitions. As you change the average jiggling rate of the molecules (heat them up) you suddenly reach a rate where the crystal isn't stable.
    – Ron Maimon
    Commented Apr 17, 2012 at 7:44

A philosopher should really, really study cognitive science / neuroscience. There are all sorts of insights we have now about how perception works, how we learn, and so on.

For example, after studying neuroscience, one realizes that introspection is an absolutely horrible guide to how perception actually works. The system is designed to present a seamless view of the world without alerting us to its inbuilt flaws and nature. This is very helpful when you're trying to dodge a lion long enough for the rest of your tribe to arrive and save you, but not so helpful when you're trying to understand the nature of perception sans neuroscience.

  • Will agree with the first paragraph. But on the second paragraph, the philosopher will say that even after studying neuroscience, we cannot be certain that the claim "introspection is an absolutely horrible guide to how perception actually works" is true.
    – Pacerier
    Commented Jul 10, 2014 at 7:47
  • @Pacerier - Well, said philosopher will either simply be pointing out that we cannot be certain that any statement about empirical matters is true (hardly news!), or said philosopher doesn't understand the evidence.
    – Rex Kerr
    Commented Jul 17, 2014 at 20:17
  • False dilemma, there are also those who understand the big gap between physics and neuroscience and how the former is more certain than the latter.
    – Pacerier
    Commented Jul 17, 2014 at 20:46
  • @Pacerier - What does it matter if some physics is even more certain (which it is)? There are oodles of psychophysics experiments that illustrate all sorts of non-introspectable aspects of perception. Introspection is a good guide to what perceptual qualia seem like (kinda tautological there). But implementation (and thus the link between stuff and perception of stuff) is not. For instance, can you introspect that there is a flash lag illusion (en.wikipedia.org/wiki/Flash_lag_illusion) or what causes it?
    – Rex Kerr
    Commented Jul 17, 2014 at 21:41

For anyone interested in the philosophy of science, I think Eugene Wigner's The Unreasonable Effectiveness of Mathematics in the Natural Sciences is an important piece, but it may be hard to truly appreciate his observations if you don't actually know physics on a technical level.


Here's philosopher Eric Steinhart's answer to the question (the math part anyways), his book More Precisely: The Math You Need to Do Philosophy. From the publisher's website:

More Precisely is a rigorous and engaging introduction to the mathematics necessary to do philosophy. Eric Steinhart provides lucid explanations of many basic mathematical concepts and sets out the most commonly used notational conventions. He also demonstrates how mathematics applies to fundamental issues in various branches of philosophy, including metaphysics, philosophy of language, epistemology, and ethics. This second edition adds a substantial section on decision and game theory, as well as a chapter on information theory and the efficient coding of information.

Here's the table of contents. And here it is in collapsed form, in case the link goes dead:

  1. Sets
  2. Relations
  3. Machines
  4. Semantics
  5. Probability
  6. Information Theory
  7. Decisions and Games
  8. From the Finite to the Infinite
  9. Bigger Infinities
  • Judging from the chapter list it is a good start.
    – Deipatrous
    Commented Sep 18, 2023 at 15:34

The short answer is "Enough not to be easily misled". And to me what that means is that one needs to be immersed in an academic culture that will allow others to care about your work and to contribute perspectives from outside your discipline.

I do not think that the philosophy of mind, for instance requires any understanding of neurophysiology. But it requires the ability to capture a metaphor that will not be derailed by idiosyncratic or purposefully misleading interpretations of neurophysiology. You need a grasp, at a man-on-the-street level of someone who might watch Horizon or Nova.

Most importantly, you need a good sense of who is making sense. To some degree, Kuhn can save you here: when it is most important, that is a sociological question, and need not be addressed by actual understanding of the field's internals.


As a biologist, I tend to refer to biology as the "central science". It builds upon physics and chemistry, and studies how we are related to other animals, including our physiology, psychology and behavior. Lack of a minimum knowledge of biology usually turns wanna-be-philosophers into endless readers of endless texts, from which they have no idea where ends the known facts and begins the hypothetical tales and religious prejudices.

Beyond that, the relation of biological sciences with a good knowledge of health could avoid, for instance, the alarming rate of obesity in many "developed" countries, or the way we are destroying all Earth's biodiversity to buy an unsustainable lifestyle. The list of important points is huge, and it astonishes me how the word "physics" appear much more in such discussion than the word "biology".

  • I think my text answers the question, and goes beyond it. The anonymous downvote could please explain itself?
    – Rodrigo
    Commented Jan 29, 2015 at 0:41
  • I didn't down vote but I suspect it has something to do with posting this answer in the philosophy exchange :p
    – Cdn_Dev
    Commented May 11, 2015 at 15:52
  • It seems that most "philosophers" in the "West" don't know how to philosophize. Most seem to think it is about reading and writing.
    – Rodrigo
    Commented May 11, 2015 at 23:50

Absolute reality = truth = fact is that which exists independent from observation and experience.

Absolute reality is that with exists objectively whether or not we are aware of it and believe it. Absolute reality exists totally seperate from anyone's capacity to observe, measure, define and/or believe it. It exists independent from any source or origin.

Philosophy is a methodology developed to contemplate the nature of absolute truth and incrementally improve our understanding thereof by first making hypotheses and validating those hypotheses with thought experiments applied to our subjective experience.

The scientific method is a refinement of philosophical methodology replacing thought experiments with empirical data and eliminating logical fallacies, making the validation mechanism far more reliable and therewith significantly reducing error.

The scientific method is thus a methodology that gradually reveals objective reality with increasing detail and accuracy and therewith also allows us see the interconnection between the different elements and attributes of objective reality that in turn helps us manipulating objective reality to our advantage. It allows us to understand how we can find happiness and safety and build the tools to fulfill them.

As a refinement of philosophical methodology, it's also worth pointing out that the scientific method is capable of adressing every issue philosophy deals with with greater accuracy and detail.

Questions the scientific method cannot answer with any degree of certainty or accuracy whatsoever are therefor also questions philosophy cannot answer with any degree of certainty or accuracy whatsoever. Abandoning the scientific method to answer questions it cannot answer at this point in time and looking for an answer in philosophy instead is therefore utterly pointless.

The vast majority of things that were mysteries to our ancestors have already conslusively been answered by science and science alone. Wherever mysteries remain, science is indefinitely the most reliable methodology to answer it. The belief that any mystery left unsolved by science can be solved or has been solved by any other means is nothing but wishful thinking and a naive response to fear of the unknown.

What about ethics?

Science both teach us a lot about how our actions influence our happiness and stability. Herein, neuropsychology typically focuses on individual behavior, whereas sociology focuses on the collective components components and biology focuses on genetic components of our behavior.

Combined, neuropsychology, sociology and biology give a rather complete picture of human behavior and human consciousness. It allows us to model human nature in a consistent way, which in turn allows us to develop a rational moral foundation based this model.

Adding to that, mathematical studies like game theory can help us determine the impact of our actions and assess that impact with greater clarity.

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    "Philosophy is a methodology developed to contemplate the nature of absolute truth [..]" -- sure, but that's only one part of it. Not all ethics fall under that narrow idea, for example.
    – user2953
    Commented May 11, 2015 at 18:03
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    The paragraphs you added don't relate to ethics in the way I hinted at. They relate to happiness and stability, not to senses of "good", "morality", "identity", etc.
    – user2953
    Commented May 11, 2015 at 18:08
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    All I'm saying is that what you say philosophy is does not cover everything what is typically considered to be philosophy. For example, Kantian ethics do not have an origin in happiness- / stability thinking.
    – user2953
    Commented May 11, 2015 at 18:18
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    Also that is one of more possible opinions. It is OK to mention such an opinion, but then you should make it clear what the premises are and that it is just one of more possible opinions.
    – user2953
    Commented May 11, 2015 at 18:24
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    @JohnSlegers, I mean this constructively, and I hope you take it as such: evidently your argumentation is not solid, because smart people disagree with you. Commented May 11, 2015 at 21:35

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