Showing posts with label maths. Show all posts
Showing posts with label maths. Show all posts

Thursday, 8 May 2008

Pinker on the applicability of Darwninan natural selection to human cognative faculties

Pinker uses Dawkins argument in The Blind Watchmaker that a complex organ that looks like it is perfectly designed for the task it performs only has one possible explanation - evolution through natural selection. In Dawkins words, "Natural selection is not just an alternative to chance. It is the only ultimate alternative ever suggested"[1] or as Pinker puts it,

Natural selection is not just a scientifically respsectable alternative to divine creation. It is the only alternative that can explain the evolution of a complex organ like the eye. The reason that the choice is so stark - God or natural selection - is that structures that can do what the eye does are extremely low-probabilty arrangements of matter" p360

However if there is a rich language organ there are also other mental 'organs'. As Chomsky states:

The language faculty is one of [the] cognitive systems. There are others. For example, our capacity to organize visual space, or to deal with abstract properties of the number system, or to comprehend and appreciate certain kinds of musical creation

If we assume UG is true then other mental computation must also have a genetic base. However, it is hard to imagine what selectional pressures led to a faculty to appreciate music or to deal with abstract number lines. These could indeed be examples of random outgrowths of a larger brain due to some unknown law of physics or computation or some concomitant co-evolution of mental organs i.e. a mental organ tagging onto another mental organ. Yet from Dawkins point of view our 'mathematical organ' is perfectly designed to deal with n+1 forever, we can comprehend to keep adding 1 to whatever number we reach, it's trivial. A parrot can't, after 7 it gets stuck. Does this mean natural selection is the only possible explanation? I don't feel it is. I have a (completely baseless) feeling that something as complicated as the human brain has to have a richer explanation than natural selection. Sure the eye is complex but it pales in comparison to the brain.

On the other hand perhaps that is why mathematics is so hard to grasp for so many. It isn't a result of natural selection and isn't properly a mental organ. Every healthy person can converse in a complex rich way. Not everyone can handle even linear equations.

Edit: Dawkins just said something relevant:

there is still controversy over the theory that natural selection is the dominant driving force. I think no body would doubt it is the dominant driving force of adaptive evolution but many people, including me, doubt it's the dominant driving force behind all evolution at the molecular level


Monday, 1 October 2007

Infinity, The Universe and Probablity

A couple years ago I thought in an infinite universe anything that has the merest possibility of occurring, will occur. So the chance of me sitting here turning into a rock may be 0.01e99 (no maths notation on blogger:(], but assuming an infinite universe and basic probability theory:

∞ * 0.01e-99= ∞

i.e. that is something (probable) rather than nothing (impossible). Thinking of it without mathematical notation we can just use the infinte Infinite monkey theorem:

The infinite monkey theorem states that a monkey hitting keys at random on a tpewriter keyboard for an infinite amount of time will almost surely type a particular chosen text, such as the complete works of William Shakespeare

I always assumed however that:

∞ * 0 = 0

Meaning that, an event that is simply impossible, will never occur because infinity multiplied by an impossible probability (0) is still zero. However, reading The God Delusion I came over Aldous Huxley's mathematical 'proof' of the existence of god

You know the formula: m over nought equals infinity, m being any positive number? Well, why not reduce the equation to a simpler form by multiplying both sides by nought? In which case, you have m equals infinity times nought. That is to say that a positive number is the product of zero and infinity. Doesn’t that demonstrate the creation of the universe by an infinite power out of nothing? [p108 GD]

I don't think this is a decent proof for the existence of god, in googling this I came across this proof that ∞ * 0 is not 1:

no one can EVER finish multiplying zero times infinity, therefore the answer will always be undefined. Even though logic dictates that the answer will never not be zero, this answer will never be reached. Therefore, trying to multiple zero times infinity is undefined.

Therefore, postulating the universe is indeed infinite (many worlds, brane worlds parallel or sequential via big bangs/crunches) then nothing, absolutely nothing, is impossible. Because the product of 0 and infinity != 0.

I don't think this is just of philosophical significance, because many current theories in cosmology postulate infinite parallel universes/branes etc.