Types of argument, and reliability.

[ceptimus]

I tend to believe that mathematical arguments (or proofs) are more reliable than scientific ones, and that scientific arguments are in turn more reliable than philosophical ones.

But my reasons for believing this are based on philosophical ideas, and this seems wrong, given that this hierarchy classifies philosophical arguments as not very reliable.

Help me out here.

[livius drusus]

I don't know about the hierarchy, ceptimus. It seems to me different approaches are more effective in different contexts, and that in fact a mathematical argument used to understand, say, human emotion wouldn't be particularly useful or reliable. (Watch you bastards come up with the perfect mathematical explanation for my daddy issues.)

I guess my point is that I don't think there's a good reason to create hierarchies of validity divorced from the nature of what is being argued or examined. Is a socket wrench more reliable than a flathead screwdriver?

[seebs]

This is one of the reasons I'm so militantly agnostic; I have no reliable way of finding out whether any of my methods are actually reliable!

[Ymir's blood]

This is one of the reasons I'm so militantly agnostic; I have no reliable way of finding out whether any of my methods are actually reliable!
From a totally sceptical point of view, nothing is reliable. All we have to go on is based on sensory data. Since we have no independent verification of our senses, every argument based on them is essentially begging the question. :thinkup:

[dave_a]

I tend to believe that mathematical arguments (or proofs) are more reliable than scientific ones, and that scientific arguments are in turn more reliable than philosophical ones.

Well sure, because mathmatics is precise, science is rigorous and demands rigid procedures and reproducibility, but still falls short of the standard of truth math provides. Philosophy is inherently less certain than either of the two because it rests upon presuppositions which themselves may not be true.

But my reasons for believing this are based on philosophical ideas, and this seems wrong, given that this hierarchy classifies philosophical arguments as not very reliable.

Help me out here.

I don't think you are wrong at all. At least I don't disagree with you.

Still, I think Liv makes a valid point in that all are tools to be used in different circumstances. While there may be math geeks around who will argue this, there are things we wish to explore, understand and know that math can't answer. When that fails we can turn to science, but there are things science cannot or hasn't yet answered. At this point we turn to faith or philosophy. In most cases philosophical thinking is far more rigorous and examined than faith.

So I would say there is (at least) mathematics, science, philosophy and faith.

While each are tools suitable for different things, I agree with you in the sense that when trying to understand something I would approach each of the 4 in the order I listed to see if it could provide me with what I was looking for. I wouldn't accept the conclusions from one if a "higher" approach yielded a different answer.

[seebs]

From a totally sceptical point of view, nothing is reliable. All we have to go on is based on sensory data. Since we have no independent verification of our senses, every argument based on them is essentially begging the question. :thinkup:

Exactly!

All I feel comfortable asserting that I "know" is that I exist, and I have experiences. I currently believe some of those experiences to be direct sensation of things external to me, and some of them to be what I call "memories", which appear to be some kind of record of previous experiences. But... I can't prove that. In fact, I have very interesting results, from dreams, of observing that I can very carefully consider the question of whether my memories are consistent, and conclude that they are, then when I wake up, realize that they were not. (e.g., books I read in a dream change their words every time I read them, but in the dream this seems normal.)

Hmm. I wonder; if I had less philosophical dreams, would I be less skeptical?

[wade-w]

I tend to believe that mathematical arguments (or proofs) are more reliable than scientific ones, and that scientific arguments are in turn more reliable than philosophical ones.

But my reasons for believing this are based on philosophical ideas, and this seems wrong, given that this hierarchy classifies philosophical arguments as not very reliable.

Help me out here.

First, exactly what do you mean by "reliable?"

In logic, we make a distinction between valid and sound arguments. An argument is considered valid if the conclusion follows from the premises, with no fallacies or error in the derivation. An argument is sound if it is valid, and the premises are known to be true. Thus an argument can be valid without being sound, and since as seebs and others have pointed out it is impossible to be completely certain if your premises really are true logicians rarely concern themselves with soundness.

Now, in mathematics we certainly have premises. IOW, certain basic statements, usually referred to as axioms or postulates, are taken to be true without proof. Vary these assumptions, and you can get wildly different results.

Take geometry, for example. What is taught in high schools as Geometry is what is called Euclidean Geometry, and is based on Euclid's Elements, a very old book. Now, this was accepted as gospel for several thousand years, until the mid-19th century. However, one of the axioms in this system never really sat well with most mathematicians. It just didn't feel like it should be an axiom; most felt it should be a theorem and just about every mathematician from Euclid until then tried prove it. But nobody ever could. So at about the same time, several people decided to assume that this axiom was false and prove that this would result in some contradiction. This would finally prove the statement as a theorem once and for all. Well, nobody ever found a contradiction, either. In fact, what was finally proven was that if the standard Euclidean Geometry was valid, then so was the alternative. And even worse, there wasn't just one alternative, there were two!

These alternate geometries, collectively called Non-Euclidean Geometry, give what sound like rather strange results. For example, the sum of the angles of a triangle is not equal to 180 degrees, as you were taught in high school. Also, there is no such thing as a rectangle (and thus no squares either). Also, the good old Pythagorean Theorem is false in Non-Euclidean Geometry. And yet Non-Euclidean geometry is just as valid as Euclidean Geometry. So which do you consider the more reliable?

[wade-w]

I didn't mean to kill this thread. Where'd you go, Ceptimus?

[Brimshack]

I wonder if it really makes sense to discuss the reliability of different types of arguments, given the fact that they are intended to be used with different types of problems. Does it really make sense to say that one type of argument is more reliable than another when in fact they may not be applicable to the same sorts of questions?

[livius drusus]

My thoughts exactly, Brim. It's the whole socket wrench vs. flathead thing: they're both reliable when used to do the jobs they're made for.

[ceptimus]

I agree with the 'right tool for the job' analogy. But I still feel that mathematical arguments stand firmer than philosophical ones.

Agreed we have to state the axioms, and by changing Euclid's fifth postulate, we get elliptic or hyperbolic geometries instead of Euclidean ones. But once we have stated our assumptions, proofs in mathematics stand firm. Only loonies continue to attempt to square the circle once it's been proved impossible.

And I know about Gödel and incompleteness and all that stuff, but that doesn't render existing proofs invalid, it merely shows that there are some mathematical things that are true, but unprovable.

Contrast this with the history of philosophy. People have been arguing about the same topics for three thousand years or more, and there is still hardly any common ground that is accepted by everyone. Different people have different interpretations of words such as 'consciousness', so philosophers can't even be sure they are understanding each other, or even arguing about the same thing, much less come to any agreement.

Some philosophical arguments do seem compelling, but never as compelling (to me anyway) as a mathematical proof.

[Clutch Munny]

It's not a good classification scheme. Math and philosophy are in many respects closer than math and "science" -- a misleading term here since science makes free use of mathematics and philosophy both; better to say something like "empirical" or just "inductive".

Math gives you certainty-relative-to-system-of-axioms. Whether there is a single such system, and whether it is at base too certain to be rationally doubted, is an important and open question. More than a few very accomplished logico-mathematical figures were/are deeply concerned with the meaning and/or legitimacy of classical mathematics, for example -- i.e., with methods of indirect proof. So runs my potted history of Brouwer, Hilbert, and Frege; anyhow Goliath is the, or a, local mathie so we might ask him whether he knows the history of these discussions.

Similar remarks apply to pure logic; and outside of pure logic there is no useful category of characteristically "philosophical" argument. Philosophers incorporate empirical evidence all the time. (Examples where they don't are easily found in history, but this reflects not a truth about philosophy but a historical truth about the generally dodgy grasp of empirical evidence.)

As with most things, there are cases and there are cases. Arguments are best evaluated by the plausibility of their premises, all things considered, and the correctness of their structure. A classification schema that lets us say, "Oh, that's an argument of kind K!" over and above the virtues of the argument itself is one that invites reliance on Refutation by Labelling and substitution of disciplinary prejudices for critical analysis.

[viscousmemories]

Clutch took the words right outta my mouth. :D

[wade-w]

It's not a good classification scheme. Math and philosophy are in many respects closer than math and "science" -- a misleading term here since science makes free use of mathematics and philosophy both; better to say something like "empirical" or just "inductive".

Math gives you certainty-relative-to-system-of-axioms. Whether there is a single such system, and whether it is at base too certain to be rationally doubted, is an important and open question. More than a few very accomplished logico-mathematical figures were/are deeply concerned with the meaning and/or legitimacy of classical mathematics, for example -- i.e., with methods of indirect proof. So runs my potted history of Brouwer, Hilbert, and Frege; anyhow Goliath is the, or a, local mathie so we might ask him whether he knows the history of these discussions.


This is basically what I was referring to earlier. The realization that there were three equally valid geometries led to some rather contentious arguments about the nature and foundations of mathematics. Brouwer championed Kroenecker's view, which was a strict constructionist one. This school of thought is also called intuitionism, and basically claims that the only valid mathematical objects are those that can be directly constructed from the whole numbers in a finite number of steps. They reject the Law of the Excluded Middle, and most forms of indirect proof.

Frege felt that mathematics could be generated from pure logic; this view was later taken up by Russell and Whitehead. Indeed, their famous Principia Mathematica was an attempt to do exactly that. This view is called Logicism.

Hilbert's school is called Formalism, and basically contends that mathematics is at root a game, played with marks on paper according to certain agreed upon rules of inference. One major focus of the formalists, however, was to find that one basic set of axioms that Clutch refers to above. Godel's work is generally accepted to show that no such axiom set is possible.

These views and various attempts to place mathematics on a more firm and rigorous foundation were hotly debated in the latter part of the 19th and early 20th centuries. As far as I know, there is no real concensus even today, though in most places mathematics is taught in a formalist manner.

[Ymir's blood]

My thoughts exactly, Brim. It's the whole socket wrench vs. flathead thing: they're both reliable when used to do the jobs they're made for.
Yes, hammering and prying, respectively. :D

[livius drusus]

Don't make me come after you with my solderer. :welder:

[Brimshack]

Hey I could use one of those.

[Clutch Munny]

Well said, wade. It's worth adding that Brouwer's challenge to the methods of classical mathematics was not a philosophical excursion launched in ignorance of what "real" mathematicians do. He was an accomplished practitioner of the methods he criticized, and did some very important work in classical topology as a means of securing a prestigious chair, whence he taught his dissenting view.

[Clutch Munny]

Don't make me come after you with my solderer. :welder:


You solder with an acetylene torch? Or is that a flame-thrower?

I suppose you use an arc welder for iron-on patches, then...

:chin:

[wade-w]

Well said, wade. It's worth adding that Brouwer's challenge to the methods of classical mathematics was not a philosophical excursion launched in ignorance of what "real" mathematicians do. He was an accomplished practitioner of the methods he criticized, and did some very important work in classical topology as a means of securing a prestigious chair, whence he taught his dissenting view.

Yes, and Kroenecker was a well known algebraist who went so far as to deny the existence of transcendental numbers. When Pi was proven to be transcendental, he said something to the effect of "That's nice, but it's meaningless, since there are no such things."