I firmly believe that philosophy without mathematically oriented thinking at the core is limiting. Teaching and studying of formal logic has been traditionally part of philosophy. In practice there is no real border between philosophy and formal sciences (logic, mathematics, statistics and theoretical computer science).

Consider the problem Kant faced in Prolegomena to Any Future Metaphysics, §13. Two objects that are intrinsically alike must be interchangeable. But there are objects that are intrinsically alike and you can't exchange them. Kant was incredible philosopher but he was thrown off by looking at his hands. Left hand and right hand seem to be intrinsically same, but you can't replace one with another. Kant concluded that things like chirality and mirror images could not be understood with intellect and reasoning using concepts. Time and space were part of sense intuition. Mathematics would disagree. Spatial intuition is not fundamental building block for reasoning about time and space. Algebra, geometry and topology are.

Many modern day problems facing humanity can be understood only as systems thinking using, probability, mechanism design, games, incentives, equilibrium, trade-off, hysteresis, etc. But many schools of philosopy still try to use tools and concepts that can't describe the system.

edit: Several important contemporary philosophers and schools of philosophy are not are in fact limiting themselves. Saul Kripke for example.

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I think philosophy should be part of curriculum in school. But it should be live philosophy centered in problem solving and rational thought. The traditional history oriented curriculum should be part of history. History of science, technology, philosophy, economics and ideas in general is more important than the narrative trough kings and power cliques.

It's very much like learning the history of mathematics without being able to e.g. solve an integral, or model a problem in the real world and make meaningful observations about the model and its relation to the subject. History is fine if it's what you're into, but the other is not taught in any widely disseminated fashion and I believe people and society are deprived as a result.

There are entire sub-disciplines of philosophy that are useful and practical for everyday decision making. Here I would disagree that you need a formal background in mathematics. Game theory, statistics and the like are certainly valuable, but they are still more a part of the engineer's or mathematician's toolkit and will be glossed over due to their rigorous technical underpinnings that are simply out of reach for many individuals. Rhetoric, critical thinking, stoicism, ethics, and the like are pretty approachable topics that have somehow been elided from the educational system in any established fashion in the US - and they used to be at the core of a liberal arts education. Great religious thinkers, politicians, and intellectuals have left a legacy of work that speaks across time and space to a modern reader.

A lot of 'continental' philosophy seems to be a mix of historical analysis, and cult-of-personality/lit-crit...

In PTAFM 13, Kant is arguing that neither space nor time are intrinsic properties of things in themselves, and basically making the point that properties like position and congruence are relational, not intrinsic. Spatial relations fit with the intuition of space, which is the form of external experience (and one should think of this as the abstract form of any space whatsoever - people tend to think that Kant is undermined by the development of non-euclidean geometries, but I think one can push the abstraction so it fits equally well). Experience of- and reasoning about spatial objects involves both intuition that corresponds with the forms of space and time (which applies to any experience whatsoever, outer or inner), and the operation of the understanding through concepts. Now, reasoning from principles which might apply to spatial objects (in the case of algebra, geometry and topology) can go through concepts alone as long as it is logical. But those principles would be meaningless for us if we didn't have experience of any spatial objects whatsoever (if we didn't have the pure intuition of space).

"Thoughts without content are empty, intuitions without concepts are blind. The understanding can intuit nothing, the senses can think nothing. Only through their unison can knowledge arise." (KrV A51).

In philosophy you get this all the time because people refuse to agree on definitions like truth, consciousness, goodness, evidence etc. So debates generally involve people talking past each other until someone gets bored. If we're lucky spectators will judge by popular acclamation which side won or lost. Participants rarely concede their position.

No, in this case he was mistaken in some basic assumptions about what Kant is doing in that text. The paragraph he mentioned starts with: "Those who cannot yet rid themselves of the notion that space and time are actual qualities inherent in things in themselves, may exercise their acumen on the following paradox." So in this case it isn't even the case that he is proposing something about Kant one can reasonably disagree on (like my suggestion that Kant is not undermined by the development of non-euclidean geometry, something which I would be willing to concede given evidence against it).

On the other hand, in my experience, actual debates on philosophy usually revolve around conceding some initial assumptions and then debating on what follows from them. Of course, one can always move from the internal questions about what follows from those assumptions to external questions about what follows if we rejected those assumptions, but this move is usually well motivated by internal conflicts. One might, for example, find some dilemma for which no option is acceptable, and thus be forced to retreat to discussing the assumptions. And it is not a given from the outset whether no such conflicts will arise.

The problem I got with philosophy is that it's supposed to be applied knowledge. But it's always consumed as an intellectual one.

As a result, the smartest philosophy lovers that I know are incredibly unhappy. They know so many things. But knowing something doesn't mean you are able to do anything about it.

Some becomes cynical. Other become depressed. Other procrastinate to hell.

But only those who actually take the knowledge and try to act on it, improving themself in the process, ends up happy. And once they get there, they usually don't quote much philosophy anymore, except for humorous purpose or make someone feel better.

Bottom line: it's interesting to look for the meaning of life. But it's necessary to actually stop and live your life unless you want to have the joy of Sartre, the energy of Rilke, the sense of purpose of Kant and drive of Schopenhauer. Hint: you really don't.

Can you elaborate on that? On why it's supposed to be applied knowledge I mean?

To me philosophy is one of the most abstract forms of thought available, and to me abstract is the antithesis of "applied". Maybe the furthest I would go is to state that philosophy is knowledge applied recursively on itself. But that already seems an abstract formulation to me.

However, if you use your knowledge of philosophy to actually do something in your life instead of trying to think your way into the abyss of infinite dissection, it becomes a powerful tool.

This is the main difference between very old philosophers (Seneca, Gautama, Epicurus...) and more recent ones (Kant, Sartre, Schopenhauer...). The first ones try actively to make something out of their thinking. They derive a way of life out of it. The second ones try to understand more deeply, categorize more things, raise more difficult questions.

In the end, by reading the active ones, you can feel the joy, the sense of purpose and solutions arising. If you read the other ones, you can almost picture them trying to kill themself.

But yeah I don't think one is better than the other. Each provide value.

I would actually claim that to live your best life, be an engineer. They seem to me at least to be living in that sweet-spot between the abstract, creative and the concrete.

http://rationalwiki.org/wiki/Salem_Hypothesis

http://scienceblogs.com/tfk/2007/11/11/the-salem-hypothesis-...

http://www.theness.com/index.php/creationists-mechanical-eng...

I'd say there's far more than anecdotal evidence, but not, obviously, not enough to draw necessarily firm conclusions. So I would say there's enough question to sustain further scientific inquiry. That, interestingly enough, puts us in a position at odds with one of a common traits noted among engineers, in psychological terms: 'need for closure'; which would put the 'engineering mindset' in conflict with the 'scientific mindset', due to the open-ended nature of the science. (Of course, this is only my own speculation.) Which is why we get things like this: http://cosmicfingerprints.com/ee/

Harder numbers:

http://www.nuff.ox.ac.uk/users/gambetta/Engineers%20of%20Jih...

(expanded book form) http://press.princeton.edu/titles/10656.html

(less numbers, but foundational)

http://www.sicotests.com/psyarticle.asp?id=212

http://www.sicotests.com/psyarticle.asp?id=235

Maybe it is a naive view on my side but I expect that additional education will make it less likely that one believes religious claims to be true in general and this extends to becoming an engineer and being a creationist.

But even if there was indeed a correlation of the form claimed by the Salem hypothesis, I would naturally want to look for traits that make it more likely for one to become an engineer and a creationist, not for something that causes engineers to become creationists.

You did not explicitly spell it out this way and I am inclined to think you do not think this causation exists, but your response to a comment suggesting that it might be a good choice to become an engineer at least allows the interpretation that becoming an engineer causes becoming a creationist.

And I obviously consider the idea of a causal relationship between being an engineer and being a creationist even more unlikely than that of certain traits increasing the likelihood of becoming an engineer as well as a creationist.

Not that it is unlikely in the general case that learning about X makes one more likely to also believe Y, that is actually certainly pretty common, but in the concrete case I am really unable to see which things one learns when becoming an engineer are suitable to turn one into a creationist.

Finally I am not sure what you wanted the express with the Evolution 2.0 article, but at least in the linked article the reasoning is heavily flawed.

[1] https://groups.google.com/forum/#!topic/talk.origins/Xunl5Sl...

Does common philosophical discourse generally attempt this though?

Most philosophers are not very careful thinkers. There is a lot of circular thinking a lot of apriori assumptions and so one. Mostly the designers who are worth listening to IMO are those who broke down previously held illusions.

I mean, howdo you explain modus ponens without the notation? You say that not A and not B is the same as not A or B. And that's a poor transpilation from math notation to words. What if you never saw the math notation and had to use regular words?

And then to spell out the proof? Oof

I was a math major for much of my undergraduate career, but by my junior year I realized that I was actually only interested in the philosophy behind it. In fact, I had little interest in my freshman Philosophy 101 course, though after studying math, I realized that I would have loved to study philosophy more. As a freshman, I was enamored by math as it was this ivory tower of abstract truth.

I loved learning about what math could say and how the notation worked and the notion of formal proof etc. but slogging through proofs was abhorrent after awhile and I dropped the program. I had come to understand that math was actually just the same sort of discourse as "soft" philosophy, just formalized. Learning about counterexamples to math as a "closed system of reasoning"[0] (Godel's theorems, constructive mathematics, etc.) simultaneously ruined my perception of the nobility of math and spurred my interest in philosophy (specifically analytic philosophy and philosophy of language) and cognitive science. That's where the real "unsolvable" or "interesting" problems lay (e.g. the mind-body problem).

I would highly recommend anyone who is intellectually curious to learn and understand formal mathematics for the purposes of understand philosophy, but also to recognize when to quit (if ever!).

[0] - I may be butchering these terms, but I hope the meaning is clear

Can you explain this? Math isn't perfect, so might as well abandon all formality? This feels like "Science doesn't have all the answers, so maybe I can find them in the bible?"

I'm not sure how constructivism hinders mathematics, or Godels in the long run.

The mind-body problem is unsolvable because it rests upon vague or false assumptions; It is philosophical nonsense : "communicating badly and then acting smug when you're misunderstood is not cleverness." https://xkcd.com/169/

Of course, a one size approach won't fit everybody.

In math you learn to follow a series of developments, building structure that leads to statements that are true. These are the foundation for more bricks in the wall and even bigger constructions. e.g. The proof of Fermat's last theorem.

My sense of philosophy - after reading maybe a dozen of the classics, was there was little that was accepted as true. Yes, there are self-consistent chains of reasoning but the foundation blocks are more a matter of "taste", and the amount of rigor in the chains varied - complicated by the fact that human language is inherently not very precise.

Please remember that the entire STEM field grew out of the questions of those philosophers you dismiss as "no clear answers": our entire scientific process (empirical theory-building) is based on the previous explorations of philosophers on truth and knowing. Similarly, both capitalism and Marxism grew out of the questions of philosophers about the structure of society.

You need questions before you can answer them. Philosophy deals with the questions. Our answers to those questions have become the main pillars of modern society.

This isn't a flaw of philosophy. Lots of fields of human study are great. Math just has some things that make it unique. But just like defining what art is, it's hard to pin down exactly what it all is.

(Or rather, if you believe in the core, you are automatically a analytic philosopher? A school I have many sympathies for, but hardly the only or even mainstream of philosophy.)

Mathematics gives you the kernel of the universe. Its a much finer grained tool that represents absolutes.

Can you look at the heat equation and see the beauty in it? https://en.wikipedia.org/wiki/Heat_equation

That the heat from your laptop flows in this way thoughout the air around it. As does the warmth of breath, or body heat or the heat surrounding an open flame?

And then there are logical principles that create "elegant" logical structures like in group theory, where you develop two different ways of looking at things only to find out they were one and the same all along (e.g. Lagrange's theorem).

Philosophy can invite discord due to the circular nature of using language to define language with the intention that this is supposed to bring clarity to our reasoning. Not to mention that the name "Philosophy" is often co-opted by charlatans to advance their brands.

But Philosophy coupled with Mathematics brings a kind of calm that's removed from language and an appreciation for simple, provable, truths.

In schools and colleges this is not always the case that we learn philosophy of a subject. Especially in schools. This is sad.

I guess we'll just have to completely disagree on that.

Is this sarcasm?

If not, can you outline "the supposed benefits of doing math" versus the benefits of studying philosophy, that are "far more applicable to most people's everyday lives"?

My question: what does graduating with a degree in philosophy say about one's abilities, and how can you show it? (I would argue the types of problems that an upper-level science/math/CS person is expected to solve very clearly shows analytical and creative ability to anyone with a little knowledge of the subject)

That's obviously not true in all cases (there are plenty of dogmatic philosophies) but in my opinion it's much more common for people to teach things like math, biology, history, etc. as a series of objective, memorized facts and formulas. Doing that with philosophy is far more difficult because so much of it is subjective and relies on criticism and analysis. In that regard, learning philosophy serves as a foundation for learning all other subjects, or learning anything in life, for that matter.

I also personally believe there's more beauty in great philosophy than in any poem or song. There are plenty of Socratic dialogues, stoic passages, and political pamphlets that still give me chills when I read them.

Chapter VI in this short book of dialogues, for example: http://www.gutenberg.org/files/17490/17490-h/17490-h.htm

Any education in a topic worth it's salt should include that once you get passed the intro courses (because the intro courses are often necessary to provide context for your analysis). Whether curricula succeed in doing so, or students engage themselves enough to do so, is another matter.

In my opinion every article about some aspect of programming should begin what an explanation of the philosophy behind it's development. Why does this exist? What problem does it solve? When should it be used/not used?

Math is a lot tougher, and so, teach you discipline and rigor much better.