I'm a student majoring in mathematics. I've taken a course in mathematical logic and a course in set theory. My problem is basically that I'm always finding philosophical concepts, for example syntax, semantics, epistemology, denotation, intuitionism, constructivism, idealism, etc., and when I look up their meaning on Wikipedia there are still more and more new concepts. So I wonder that maybe instead I should start studying philosophy in a consistent and organized way instead of learning isolated concepts, but I don't know how to do that.

To be more precise, in mathematics and in every other science mostly the basic concepts are given at the beginning and then everything goes from there. This we can call it "the foundations". For example, in mathematics we can start with logic and set theory. From them every other branch is derived: geometry, analysis, algebra,..., all of mathematics. This makes its study very pleasant and organized. My question is then if there's something similar to study philosophy. Where do I start? Is there an organized way to study philosophy?

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    In the 20th century philosophy of language became the "first philosophy". You may find Miller's Philosophy of Language a lucid introduction books.google.com/books/about/… It starts with Frege introducing syntax, semantics, denotation, conotation, sense, reference, and proceeds through Russell, logical positivists, Wittgenstein, Quine, etc. criticizing, modifying, and bringing in new topics related to epistemology, realism, idealism, etc. It makes more sense when the concepts are presented in their natural logical and historical formation.
    – Conifold
    Commented Oct 29, 2015 at 0:39
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    A good modern approach is the LessWrong Sequences
    – spraff
    Commented Oct 29, 2015 at 12:00
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    One potentially problematic assumption in your question (if it is indeed an assumption you're making) is that there is anything that can be uncontroversially described as the foundations of mathematics (or any science, for that matter). There are many competing foundations. Though set theory is the most popular choice, this is (arguably) largely a historical accident. Category Theory (and, I'm told, Homotopy Type Theory) could equally well serve as the foundations of mathematics. Similar considerations apply to just any scientific theory I can think of (e.g., "the" foundations of physics).
    – Dennis
    Commented Oct 30, 2015 at 1:21
  • @Daniela, you might find interesting "A Paper on the Difficulties of Researching in the Humanities"—written by someone who researched and laid out some highly interesting and very precisely defined axioms of life (though these are not covered in this essay). lronhubbard.org/articles-and-essays/…
    – Wildcard
    Commented Oct 30, 2015 at 2:48
  • You think therefore you are a philosopher.
    – Eldho
    Commented Nov 5, 2015 at 18:13

7 Answers 7


This is very good and possibly tragic question, and I am not the best one to answer it. But the short answer is: No.

There is no faster "royal road," as Euclid first phrased it, to either mathematics or philosophy. Unlike the natural sciences, philosophy never erases its history. Thales and Heraclitus are still studied. Unlike mathematics there is no illusion of axiomatic closure and no obvious place to begin, except historically.

So, one way to start is at the beginning. Whatever your field, it is always good to study some Plato. In your case, if you take a course you may want to look at his relationship with the Pythagoreans. You might also look at some related mathematical history. For example, the way in which Greek geometrical proofs using only straight edge, compass, and hand movements. No numbers. Do not neglect the history of your field, which does intersect with ancient philosophy.

Or you could go in the other direction.

Modern philosophy begins with Descartes, who, as you may know, was the first to link geometry and algebra. It continues up to Kant. And here you must be alerted to a fork in the road, referred to in philosophy as the "Continental" and "Analytic" traditions, and commencing roughly with Husserl and Frege.

The "Analytic" tradition leans far more heavily on mathematics and logic. So this may be more in line with your interests. I would say that its central concern is how mathematics, whatever that is, links to logic and then language. But be aware, if you read or take a course in recent "analytical" philosophy about, say, logic, possible worlds, computing, AI, and such, that's fine, but you are not engaging with philosophy per se, from Thales and Plato all the way through to Kant, Husserl, Hegel, Heidegger, Sartre, etc.

In the "Continental" tradition, on the other hand, "phenomenology" and "hermeneutics" hold sway. Or did...this old divide is now regularly breached. The common ground between both traditions is, to simplify, the meaning and structure of "language." And my guess is that you went into math because you perhaps subconsciously find "language" frustrating. Very understandable, very "Analytic."

One good thing about analytical philosophy is that the terminology does not proliferate quite as rapidly as in the "Continental" tradition, where each philosopher tends to reconstruct his/her own terminology... very annoying, but there is usually a good reason. Language is real, unavoidable, and not axiomatic.

Assuming you are still in school, I would make three suggestions.

First, just take a beginning in philosophy course or a course on Plato, if you haven't already. Two, get a couple of books on the history of mathematics, which leads back into philosophy. Three, just invest in a good dictionary of philosophical terms so you can look up "epistemology"and "constructivism" and all those other terms. It just takes a bit of time and usage, like learning in French.

If you find you have a taste for it, perhaps take a course on Descartes. If you love mathematics, you really should get acquainted with the old, frustrating "spouse of mathematics," philosophy.

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    +1 I very much like the idea that the best place to start for the foundations of philosophy is yourself.
    – Cort Ammon
    Commented Oct 29, 2015 at 1:27
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    The short answer to "What are the foundations of philosophy?" is "No"?
    – user17596
    Commented Oct 29, 2015 at 13:41
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    I'm referring to the last line, gist of the question, and the idea of some "correct" or "best" way to approach study of philosophy. It's true, I often forget about the headline posting. Commented Oct 29, 2015 at 14:38
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    +1: history is always important; the past is our only resource in constructing the future - to paraphrase Simone Weil badly. Commented Oct 29, 2015 at 17:56
  • Plato is important for the history of philosophy, but many of his ideas and arguments were just terrible, and I think philosophy was held back by taking him too seriously. Plato is to philosophy as epicycles is to Astronomy. He's more useful for what his mistakes could teach us.
    – R. Barzell
    Commented Oct 31, 2015 at 16:23

As far as I know no textbook for philosophy exists alike to textbooks in the different branches of science. Probably the reason is the lack of a baseline of general accepted philosophical results.

I would recommend to start not with single philosophical terms but with reading some of the most quoted original sources:

  • Plato: Apology, Criton, Phaedo, Symposion, The Republic (Book 1 and the parabola of the cave from book 7 and the parabola of the sun from book 6)
  • Aristotle: Metaphysics, book 1 and 2
  • Thomas Aquinas: Summa theologiae (The first quaestio)
  • Descartes: Meditations
  • Hume: A Treatise on Human Nature
  • Kant: An Answer to the Question: What Is Enlightenment?, Critique of Pure Reason (Second introduction)
  • Nietzsche: Thus Spoke Zarathustra
  • Quine: Two dogmas of empiricism
  • Popper: Objective Knowledge: An Evolutionary Approach

All can be read without studying the secondary sources before. But I would recommend to inform oneself about some biographical data of the authors.

These works make a programm for about one year.

In my experience it takes several years to get one's own philosophical stance. But it's worth to aim at this goal.

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    This is a very good program that's roughly similar to the required readings when I was teaching required courses in the US.
    – virmaior
    Commented Oct 29, 2015 at 4:35

Various philosophers have tried in the past to develop a kind of axiomatic, ground-up approach to philosophy, in which the terms are neatly set out and defined and then everything is supposed to proceed deductively from there. In the early 20th century, logical positivism might well be considered such an attempt.

Eventually such attempts result in other philosophers challenging the fundamental presuppositions of the approach and there doesn't seem to be any obvious convergence towards a consensus.

A personal view is that philosophy is a kind of conceptual engineering. Philosophers address problems that are so difficult that we don't start with any well-defined concepts that will allow a certain solution. Then we chip away at a problem and try to break it into manageable pieces and analyse the pieces and try to build paradigms and models that provide insight. All branches of science start with vague concepts and generally only progress towards precisely defined terms when they are mature and successful.

In the case of philosophy, when it is successful it tends to give birth to a new branch of study. In the time of the ancient Greeks all knowledge was considered philosophy. Once philosophers such as Francis Bacon worked out the basics of the scientific method and showed how all kinds of problems can be solved by empirical study, natural science was born. Even as late as the 18th century, the terms natural science and natural philosophy were interchangeable. Later philosophy gave birth to psychology, linguistics, formal logic, political science, economics, etc. What we are left with in philosophy are the problems we still haven't solved.

So to answer your question, there is no single organized way to approach philosophy. Having said that, when it comes to philosophy in the analytical tradition, logic has a primary place. Since you are a mathematics student, your grasp of formal logic is probably already strong. You would likely find it useful to study what is often called philosophical logic, or philosophy of logic. Two good books in that space are Grayling's Introduction to Philosophical Logic and Susan Haack's Philosophy of Logics.


My problem is basically that I'm always finding philosophical concepts, for example syntax, semantics, epistemology, denotation, intuitionism, constructivism, idealism

But some of the things you've mentioned aren't philosophical. For example, syntax is a completely well-defined term for the allowed strings of symbols drawn from some specified alphabet. Semantics too is defined as the interpretation of strings of symbols, which means that there are many possible semantics of the same string. Interpretations are defined in a meta-system. Intuitionism is too vague to be a concept unless more carefully specified, since some people who claim to be intuitionistic purposely insist that intuitionism cannot be defined! Intuitionistic logic, however, is a very well-defined formal system with inference rules that make it strictly weaker than classical logic. Constructivism in mathematics is another vague concept usually used as a broad umbrella for opinions that roughly speaking want to have explicit constructions of mathematical objects before one is entitled to assert that they really exist. Some types of constructivism are objectively defined.

Nevertheless, it is true that at the bottom of the foundations of mathematics lies philosophy, because ultimately there are concepts that cannot be defined, and have to be understood even before any axiomatization. You can take a look at this for a brief explanation. Suffice to say that if you understand "if" and counting upwards from 0, then that is enough to build formal systems and use them to capture all our intended higher mathematical concepts that can be described unambiguously. But if you don't, or you disagree with that, then I'm afraid that it's impossible to put forth any solid argument at all, since conveying arguments (of any sort) require both parties to accept some kind of syntax rules, and syntax requires conditionals and symbolic strings. Without a formal system in place, every argument can be argued to be unfounded.

Also, This Stanford Encyclopedia of Philosophy article is an excellent concise reference to start looking up all these things in the philosophy of mathematics.


As others have mentioned, it is impossible to disentangle philosophy from the history of philosophy. To make things even more difficult, it also impossible to disentangle philosophy course from the philosopher giving the course: An analytic philosopher will give and entirely different course/text book form a continental philosopher, even though they are covering the exact same material. A materialist and a dualist will teach philosophy of mind in entirely different ways. That being said, there is hope.

To be more precise, in mathematics and in every other science mostly the basic concepts are given at the beginning and then every thing goes from there.

The equivalent of 'basic concepts' in philosophy is central questions. I have found the most useful text books and lectures to be organized around these central questions, and the various sub disciplines of philosophy they lead to, as opposed to the historical narrative that most courses take.

Some of these central questions/topics:

What is philosophy?


  • What is the basis for morality?
  • How do we determine right from wrong?
  • How should a person behave?

Metaphysics & philosophy of mind:

  • Do universals exist?
  • Monism vs Dualism
  • The mind-body problem
  • Freewill vs determinism
  • Ontology
  • Possible worlds vs the existing world

Epistemology & philosophy of science:

  • What is the definition of knowledge?
  • Can we be certain of anything?
  • Rationalism vs empiricism
  • The problem of induction
  • The demarcation problem: What is science?
  • Observation vs theory: When do I know that something is missing from my theory and when do I know that something is missing from my observations?
  • Bayesian vs frequentist statistics and the meaning of probability

Logic & language

  • Are arguments sound? valid? Are they fallacies
  • Aristotelean/Boolean logic vs Non-classical logics

Philosophy of Religion

  • Arguments for and against God's existence
  • The problem of evil
  • Revelations and Miracles

Political, economic philosophy and philosophy of history:

  • Modes of government: Democracy, Monarchy, etc...
  • Human rights
  • Marxism vs Capitalism

This list is obviously not exhaustive, and several of these questions and topics are interrelated. For example questions of metaphysics and philosophy or religion tend to go hand in hand, as do questions of ethics and political philosophy.

As I mentioned earlier, I have found it much easier to approach philosophy via these fundamental questions then by start with philosopher X and ending with philosopher Y.


As has been pointed out, the discussion on the proper foundations of philosophy is itself a vivid topic in the field. Many of the most studied texts in philosophy are proposals or refusals of such foundations. So, understanding and evaluating these proposals is already a task demanding a familiarity with philosophy. Where to start?

Like in mathematics, a philosophical text is not an arbitrary (as in, e.g., poetically pleasing) collection of words, nor is it a collection of what an author merely thinks to be the case. Instead, it has an argumentative structure, consisting, among others, of definitions, examples, theses, arguments (which themselves consists of premises, axioms of logic and conclusions). Argumentation is central ("fundamental") to philosophy in a way comparable to the notion of a proof in mathematics (with an obvious difference concerning their acceptance among professionals).

Understanding specific positions in philosophy is therefore not limited to understanding the meaning of its theory (or, more colloquially, "what reality would be like if the theory was true"). The more demanding, but also more fun, part is understanding why an author thinks a theory to be true and how they try to prove it. From this point of view that puts forward minimalistic, rather abstract "foundations" of philosophy, it is up to the reader's interests which philosophical texts to study -- but wherever their choice might lead them, students will do very poor without a good understanding of argumentative reasoning in general and arguments typical to philosophy in particular.

Over the centuries, philosophers have used certain ways of proving their theories. One beautiful example is the Transcendental Argument Structure, closely associated with Immanuel Kant (who, in his 1787 Critique of Pure Reason, gave theories on how it is possible humans do math and physics, and proposed an answer why metaphysics could, as a matter of principle, never arrive at a comparable level of certainty). But you will also encounter more familiar reasoning, like the reductio ad absurdum (which is widely used in math), or the notions of contraposition and supervenience.

This answer is not complete and has faults and questionable propositions, but if you were (partially) convinced of this approach to study philsophy systematically, your choice of reading and/or university courses might contain (those are from the reading list of an intro class I attended):

  • Any widespread, and beginner-friendly, introduction to formal logic. Some are designed specifically for philosophy classes (like Quine's or Graham Priest's)
  • An accompanying intro to informal logic, argumentation theory, and the use of arguments in philosophy in particular, such as those by
  • Douglas Walton ("Informal Logic - A Handbook for Critical Argumentation" and "Argument Structure - A Pragmatic Theory")
  • Nicholas Rescher ("Philosophical Reasoning - A Study in the Methodology of Philosophizing")
  • Stephen Toumin ("The Uses of Argument")
  • There is a comprehensive collection of famous philosophical arguments edited by Michael Bruce and Steven Barbone ("Just the arguments. 100 of the most important arguments in Western philosophy"). It covers a wide range of topics and has a number of fascinating examples, but the quality of the logical reconstrutions also varies widely.

I remember this predicament!

You might try looking at it a different way.

Mathematics and philosophy have, I think, two fundamental Big Ideas in common: inquiry (aka problem solving?) and convention-based propagation of ideas (in such a way that any person in the know can follow the thought process of another—though whether we can assume this is possible or happening is another question :) ). You could think of these as the fundamental skillset.

The study of philosophy is inevitably a study of its history, and, in particular for an English speaker, the history of Western philosophy. I agree wholeheartedly with a previous poster that the history of mathematics is revealing and essential (and undertaught). Following from inquiry (and informing communication of your ideas) is the skill of fostering a rich shared context through the study of history and contemporaneous thought.

In addition to historical study—which my bones nevertheless will not quite let me call fundamental—I suggest looking for and questioning the fundamental assumptions of mathematics (you have probably already begun). Consider non-contradiction, induction, and the "existence" (or reification) of mathematical objects, for example (or Alexander S King's "central questions" above—great answer!)—or, crucially, whatever has driven you to your original inquiry—then try conveying your ideas. This would be engaging in the foundation skills of philosophy.

Of course I'm implying that the "pleasant and organized" approach is suspect (as I think you might have been). If you're looking for the same kind of packaging, double major in Philosophy! You could examine a few university philosophy curricula and follow one on your own. Or, you could organize your study according to your own inquiry. Try deferring your frustration with the dendritic approach. See where it leads—you may get everywhere faster (or more thoroughly) with a "hierarchy" where you are the foundation. Yours is a good and important question. You may be looking for your own answer.

My answer: the skills of 1) inquiry and 2) the use of conventions of logic and language. Secondarily (though no less importantly): 3) fostering a rich shared context by studying the history of philosophy (thereby also giving structure to the lexicon).

(And, great fun: Shapiro's Oxford Handbook of Philosophy of Mathematics and Logic)

(And, quick and dirty: History of Philosophy without any gaps)

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