For example, many of the moral dilemmas that get tossed around as interesting become trivial to answer once you give a mathematically rigorous definition to words like "good", "moral", "utilitarian" or whatever.

Yes, but the while the conclusions become trivial, actually coming up with those rigorous definitions is not.

I'm not going to provide examples because they are likely to start a flamewar. But if you search for words like "normative" or "core principle" in my comment history, you can see how difficult it is to actually come up with those rigorous definitions. The problem is that when you do, you either come up with unsatisfying arbitrary definitions, or else you wind up implying peripherally related conclusions you probably dislike.

Yes, I am definitively guilty of being "highly mathematical" (and also highly economical and philosophical), and it pushes me to recognize incoherence in mainstream "thought".

> I usually take the presence of any proofs in my code as the cue to question my definitions.

http://stackoverflow.com/a/13241158/46571

What you sometimes see among philosophers is that even definitions that are the result of many iterations are still treated more like rules of thumb than precise definitions. Or they at least acknowledge that their definition is probably flawed, and proceed carefully.

Some proceed as if they are doing math.

I really don't understand this. You give a counterexample to a theorem; you don't give one to a definition. If someone disagrees with the definition and wants to use a different one, then they are talking about a different thing (even if they want to use the same English word to refer to each of them) and their conclusions can not be compared in any meaningful way.

Imo, you can't "proceed carefully" with a definition that is open to interpretation. If you could do so safely, then you know enough about how people could interpret your definition to form a more rigorous definition.

  >> that somebody won't be able to
  >> disagree with by counter example

  > I really don't understand this.
  > You give a counterexample to a
  > theorem; you don't give one to
  > a definition.

The classic example is "connected" from topology. Speaking very loosely, a set is "disconnected" is there is a "disconnection", which is a separation of the set into two pieces which are contained in disjoint open sets. Basically, the set is disconnected if it's made up of two (or more) pieces that can be divided by a "surface". A set is "connected" if there is no disconnection.

The problem is that there are sets we want to think of as not connected, and yet which satisfy the definition of connected as given above. Here's an example:

  X = { (x,sin(1/x)) : x in R, x>0 } u { (0,y) : -1 < y < 1 }

If that example had been thought of earlier, it's possible that the definition of "connected" might have been fixed earlier. So in a sense, X is a counter-example to the definition of connected.

Further reading:

http://planning.cs.uiuc.edu/node140.html

https://en.wikipedia.org/wiki/Connected_space

https://en.wikipedia.org/wiki/Connected_space#Path_connected...

Mathematical definitions are usually abstractions of concrete observations, but they aren't the observations themselves. This is why coming up with a mathematical definition often requires more work than coming up with any other kind of definition.

> What is "1"? The idea of "1" can only be understood in terms of experience

1 is very rigorously defined: As a natural number, it's the successor of 0. As an integer, rational, real or complex number, it's the result of mapping the natural number 1 into Z, Q, R or C in a suitable way (compatible with the semiring structure of N).

That's all nice and dandy for mathematicians, but such is still meaningless (from an overall perspective) because ultimately you end up with recursive definitions. "The successor" of "0" means nothing because "successor" isn't defined, unless you define it via dimensions (or something else). Once you try to define dimension or otherwise, you have to define dimension, which leads to defining counting, which leads back to defining 1. Appealing to experience is the only way to end the cycle.

That said, it's perfectly fine to talk about a system within itself, but it's applicability to anything real (which is what I'm concerned with - my apologies if that's not where you're going) is limited to whatever can be described in terms of reality.

The existence of a successor function is postulated, because it's an axiom of the theory of natural numbers. It's up to individual models of the theory to define this function concretely.

Fuzzy premise -> precise logic -> fuzzy conclusion

where each of those arrows leaves so much room for interpretation that you can build literally an entire subfield by arguing back and forth about what the most reasonable mapping from fuzzy to precise and back again might be, even if everyone agrees that the manipulations in between are rigorously correct.

But in general when I read philosophers disucssing social matters, I feel they are being too rigorous. Trying to cover all bases when the fuzzy nature of the subject is always going to make their abstractions leak.

They have to do this because they compete with other philosophers who will pick the nits.

Perhaps it depends on what you mean by "precise", but philosophical arguments do tend to start by defining one's terms or disputing the opponent's definitions, expressed or implied. When continental philosophers talk, that's the only part I can follow.