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I strongly disagree with the claim that philosophers fit this description. As a mathematician, it often seems to me that 90% of philosophy is spent arguing abou

I strongly disagree with the claim that philosophers fit this description. As a mathematician, it often seems to me that 90% of philosophy is spent arguing about things with extremely loose definitions. As a result, you can argue from the same starting point and come to completely different conclusions (as separate philosophers often do), because the starting point was already self-contradictory for some interpretations of the wording. As soon as you clearly define what you are talking about the problem usually becomes trivial. 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.

nercht12 10y ago
Define "loose". Edit: Jokes aside, nothing is really "strictly-defined" as we would like it to be since definitions (even the most rudimentary ones) are based on -what we might call- statistical sets of observations. We come to the conclusion that an apple is a round object because we've seen an apple from multiple angles. Math accepts axioms, but those axioms are based on accepting definitions that are rudimentary observations. What is "1"? The idea of "1" can only be understood in terms of experience - hardly "rigorously defined".
- catnaroek 10y ago
  > definitions (even the most rudimentary ones) are based on -what we might call- statistical sets of observations 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).
  - nercht12 10y ago
    > 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. ... 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.
    - catnaroek 10y ago
      > "The successor" of "0" means nothing because "successor" isn't defined 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.
redipiny 10y ago
One of the problems of philosophical discourse, imo, is that it seems to be impossible to give mathematically rigorous definitions of concepts like 'good' or 'moral' or even 'knowledge' that somebody won't be able to disagree with by counter example. 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.
- unfamiliar 10y ago
  > 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. 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.
  - CarolineW 10y ago
    >> 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. In a sense you do. Often when we are giving a definition we are intending to capture a particular idea. Sometimes the definition we give captures too little, or too much. If you find that out early enough then you can change your definition to better match what you intend. 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 } So the net result is that we have the two definitions: "connected" and "pathwise connected." 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 http://planning.cs.uiuc.edu/node140.html https://en.wikipedia.org/wiki/Connected_space https://en.wikipedia.org/wiki/Connected_space https://en.wikipedia.org/wiki/Connected_space#Path_connectedness https://en.wikipedia.org/wiki/Connected_space#Path_connected...
yummyfajitas 10y ago
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".
- SatvikBeri 10y ago
  Even in Math, coming up with good definitions is often the hard part. For example, modern point-set topology is both extremely powerful and consists of theorems that seem obvious to prove, but that's because the definition of a topology is so well-chosen.
  - catnaroek 10y ago
    I like how Conor McBride put it: > I usually take the presence of any proofs in my code as the cue to question my definitions. http://stackoverflow.com/a/13241158/46571 http://stackoverflow.com/a/13241158/46571
JohnStrange 10y ago
I'm a philosopher and agree. There are some philosophers who make sufficiently precise definitions and use reasonable formal method - where certainly not all formal philosophy is useful, though. They are in a small minority. I'd even give the same ad hoc figures, it's about 90% trash, 10% worth reading and then a few gems, so you have to be careful which authors you read and what topics to study in philosophy.

nercht12 10y ago

Define "loose". Edit: Jokes aside, nothing is really "strictly-defined" as we would like it to be since definitions (even the most rudimentary ones) are based on -what we might call- statistical sets of observations. We come to the conclusion that an apple is a round object because we've seen an apple from multiple angles. Math accepts axioms, but those axioms are based on accepting definitions that are rudimentary observations. What is "1"? The idea of "1" can only be understood in terms of experience - hardly "rigorously defined".

catnaroek 10y ago
> definitions (even the most rudimentary ones) are based on -what we might call- statistical sets of observations 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).
- nercht12 10y ago
  > 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. ... 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.
  - catnaroek 10y ago
    > "The successor" of "0" means nothing because "successor" isn't defined 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.

redipiny 10y ago

One of the problems of philosophical discourse, imo, is that it seems to be impossible to give mathematically rigorous definitions of concepts like 'good' or 'moral' or even 'knowledge' that somebody won't be able to disagree with by counter example. 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.

unfamiliar 10y ago
> 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. 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.
- CarolineW 10y ago
  >> 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. In a sense you do. Often when we are giving a definition we are intending to capture a particular idea. Sometimes the definition we give captures too little, or too much. If you find that out early enough then you can change your definition to better match what you intend. 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 } So the net result is that we have the two definitions: "connected" and "pathwise connected." 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 http://planning.cs.uiuc.edu/node140.html https://en.wikipedia.org/wiki/Connected_space https://en.wikipedia.org/wiki/Connected_space https://en.wikipedia.org/wiki/Connected_space#Path_connectedness https://en.wikipedia.org/wiki/Connected_space#Path_connected...

yummyfajitas 10y ago

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".

SatvikBeri 10y ago
Even in Math, coming up with good definitions is often the hard part. For example, modern point-set topology is both extremely powerful and consists of theorems that seem obvious to prove, but that's because the definition of a topology is so well-chosen.
- catnaroek 10y ago
  I like how Conor McBride put it: > I usually take the presence of any proofs in my code as the cue to question my definitions. http://stackoverflow.com/a/13241158/46571 http://stackoverflow.com/a/13241158/46571

JohnStrange 10y ago

I'm a philosopher and agree. There are some philosophers who make sufficiently precise definitions and use reasonable formal method - where certainly not all formal philosophy is useful, though. They are in a small minority. I'd even give the same ad hoc figures, it's about 90% trash, 10% worth reading and then a few gems, so you have to be careful which authors you read and what topics to study in philosophy.