It is usually assumed that philosophy is a search for truth, or in the (post-)(post-)-modern view where the word truth should not be uttered, that philosophy creates discourses and thereby shapes reality.
Philosophy has converged towards a subject consisting of arguments (or some kind of more or less specified manipulation of texts and meanings, as in deconstruction) backed by whatever authoritative world-view is prevalent at the moment, be it religion, science, marxism, enlightenment theory, or something else. The effect is that philosophical positions about which there is little room for arguing tend to be abandoned by contemporary philosophers. One can not make a living from finding an insight which is hard to connect with the basis for one's funding and position. This tendency has now in many cases turned into a profoundly materialistic world-view, reflected in many parts of the philosophical debate. The rejection of all possible scenarios which are not directly amenable to logical scrutiny, to political debate, or to scientific inquiry, is a choice made on pragmatical grounds. There is nothing necessary (or even plausible, given what we know from history) about the thesis that ideas which today cannot be thoroughly defended or criticized, are not true. It only illustrates how ingrained the view that we create "stories of reality" by arguments, has become. Indeed, it a realistic and plausible assumption that there are many aspects of reality which are simply too far removed from our daily politics, physical or biological sciences, or indeed Western philosophical discourses, to be efficiently approached and understood within those systems.
The Western idea that truth must be debatable, analyzable, formalizable, is certainly efficient in many, mostly practical, areas of human existence. At the same time, there is no ground whatsoever for assuming that this specific perspective encompasses all of reality. (If this is how one thinks all of reality is brought into being via various discourses, one can just as well withdraw into a solipsistic slumber, rather that reading this article.)
The methods and perspectives of philosophy have thus become biased towards those opinions on which many papers can be written and which allow for the inclusion of buzz words like cognitive science, complexity theory, or neuronal consciousness. Consequently, views which suggest another perspective are duly rejected, not because they have been falsified, but because they have become less popular and profitable. The mistake made is that the new ideas are not developed as a complement in addition to previous classical perspectives, but in order to give themselves greater power and credibility they overthrow anything in the "tradition" and put themselves on top. This practice is very far removed from the open-minded honest search for truth, and needless to say, it it very likely that this approach omits large and significant parts of reality.
Saturday 18 October 2008
Friday 17 October 2008
Against the anti-platonists
Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being (WMCF) by Lakoff and Núñez, is an attempt to explain how we understand mathematics, using principles developped in cognitive theory. This is a perfectly worthy and reasonable goal. They do however not stop there, but try to use their ideas to argue that mathematical platonism is impossible, a romantical mistake.
Let's take a look at a thing they say in the book (paragraph below is copied from Wikipedia):
WMCF (p. 151) includes the following example of what the authors term "metaphorical ambiguity." Take the set A={{},{,{}}}. Then recall two bits of standard elementary set theory:
The recursive construction of the ordinal natural numbers, whereby 0 is , and n is n-1 {n-1}.
The ordered pair (a,b), defined as {{a},{a,b}}.
By (1), A is the set {1,2}. But (1) and (2) together say that A is also the ordered pair (0,1). Both statements cannot be correct; the ordered pair (0,1) and the unordered pair {1,2} are fully distinct concepts. Lakoff and Johnson (1999) term this situation "metaphorically ambiguous." This simple example calls into question any Platonistic foundations for mathematics.
There are several clear errors here. The claim that two statements cannot both be correct because they refer to "fully distinct concepts" is not only a vague statement with no backing, it also assumes that the two instances above of the number 2 have nothing to do with each other. This is manifestly not the case, since going through the simple definitions above give a direct way of passing from one instance to the other. As an example, the fact that the two expressions
e^{\pi i} and -1 seem to be "fully distinct concepts", does not prevent them from being equal as real numbers.
The "metaphorically ambiguous" situations are precisely cases where we make connections not through intuition or metaphor, but by logic. This shows that part of mathematics consists of establishing truths which are not obviously grounded in anything in our experience as embodied beings, but rather emerge from the inner logical structure of mathematics itself. These situations are therefore examples whose existence Lakoff's and Núñez's theory fails to explain.
Therefore, rather than calling Platonism into question, these examples only serve to affirm the fact that analogy based in physical reality does not account for all aspects of mathematics.
Such careless arguments against platonism shows that Lakoff and Núñez, rather than being unbiased scientists searching for the truth whatever it may be, are on a crusade to defeat competing philosophies in order to promote their own. Mathematicians know that while these ideas are very interesting for mathematical education and in understanding how we think about mathematics, it basically says nothing about the ontological status of mathematics itself. It can at best support a viewpoint which takes as its basic principle the human understanding of mathematics, but it can never determine what the nature of mathematical objects is. This makes the theory completely consistent with platonism.
Of course this is all part of the contemporary trend in philosophy to reject platonism, dualism, and whatever philosophical theory there might be which has had a great influence for a long time. The more support a classical theory has had through the ages, the bigger the victory for those who manage to dethrone it. History has shown that such ambitions usually do not lead us closer to understanding and consensus about the world.
Let's take a look at a thing they say in the book (paragraph below is copied from Wikipedia):
WMCF (p. 151) includes the following example of what the authors term "metaphorical ambiguity." Take the set A={{},{,{}}}. Then recall two bits of standard elementary set theory:
The recursive construction of the ordinal natural numbers, whereby 0 is , and n is n-1 {n-1}.
The ordered pair (a,b), defined as {{a},{a,b}}.
By (1), A is the set {1,2}. But (1) and (2) together say that A is also the ordered pair (0,1). Both statements cannot be correct; the ordered pair (0,1) and the unordered pair {1,2} are fully distinct concepts. Lakoff and Johnson (1999) term this situation "metaphorically ambiguous." This simple example calls into question any Platonistic foundations for mathematics.
There are several clear errors here. The claim that two statements cannot both be correct because they refer to "fully distinct concepts" is not only a vague statement with no backing, it also assumes that the two instances above of the number 2 have nothing to do with each other. This is manifestly not the case, since going through the simple definitions above give a direct way of passing from one instance to the other. As an example, the fact that the two expressions
e^{\pi i} and -1 seem to be "fully distinct concepts", does not prevent them from being equal as real numbers.
The "metaphorically ambiguous" situations are precisely cases where we make connections not through intuition or metaphor, but by logic. This shows that part of mathematics consists of establishing truths which are not obviously grounded in anything in our experience as embodied beings, but rather emerge from the inner logical structure of mathematics itself. These situations are therefore examples whose existence Lakoff's and Núñez's theory fails to explain.
Therefore, rather than calling Platonism into question, these examples only serve to affirm the fact that analogy based in physical reality does not account for all aspects of mathematics.
Such careless arguments against platonism shows that Lakoff and Núñez, rather than being unbiased scientists searching for the truth whatever it may be, are on a crusade to defeat competing philosophies in order to promote their own. Mathematicians know that while these ideas are very interesting for mathematical education and in understanding how we think about mathematics, it basically says nothing about the ontological status of mathematics itself. It can at best support a viewpoint which takes as its basic principle the human understanding of mathematics, but it can never determine what the nature of mathematical objects is. This makes the theory completely consistent with platonism.
Of course this is all part of the contemporary trend in philosophy to reject platonism, dualism, and whatever philosophical theory there might be which has had a great influence for a long time. The more support a classical theory has had through the ages, the bigger the victory for those who manage to dethrone it. History has shown that such ambitions usually do not lead us closer to understanding and consensus about the world.
Thinking and dancing
Thinking is the mind dancing. Thinking in general is the mind moving in directions randomly, according to its own will, or according to physical circumstances. Thinking about pure mathematics is the mind dancing to music.
Conversely, dancing is pure expression of the body, which is to the body what pure thought is to the mind.
Conversely, dancing is pure expression of the body, which is to the body what pure thought is to the mind.
Music and Maths
All music to some extent, but especially techno music, is mathematics plus time. This echoes the classical realization of Leibniz that the pleasure the human mind derives from music is the pleasure of counting uncousciously.
Thursday 9 October 2008
Purpose
This blog is to collect my thoughts and opinions on all matters which I find worthwhile to write down. This means that the content will appear fragmented, brash, sometimes taken out of context, and will mostly (but not exclusively) be concerned with philosophy and mathematics. I like to use a minimum of words to express a maximum of meaning, and history has shown that texts written in this style tend to keep their readership better throughout time, compared to encyclopaedic treatises (examples of the fragmentary style include The Bible, Pascal's Pensees, Wittgenstein's Tractatus, Nietzsche, and the encyclopeadic - diplodocus genre includes Diderot's Encyclopedie, the Bourbaki volumes, Russell-Whitehead's Principia Mathematica. One could however argue that the encyplopaedic tomes shape the consciousness of populations without actually having to be read or re-read by newer generations, that they are (what Wittgenstein ironically enough claimed his book was) ladders that can be thrown away after they have been climbed. Such texts may thus serve a purpose, although they usually have to be written by several people with an enormous investment of labour. They are therefore unsuitable for the current format (and perhaps our current times).
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