"... intuitionistic mathematics is an essentially languageless activity of the mind having its origin in the perception of a move of time. This perception of a move of time may be described as the falling apart of a life moment into two distinct things, one of which gives way to the other, but is retained by memory."

Luitzen Brouwer

I can imagine somebody counting, beginning with 1 and then creating (more and more of) the natural numbers. But I cannot imagine the reverse process, somebody beginning with all of the natural numbers and then taking away one after the other.
This is the main motivation for intuitionism in mathematics.

The platonistic view assumes that the natural numbers are "just there" as described by Peano's axioms. I guess platonism is the majority view among mathematicians.

Perhaps this division mirrors the two different approaches I mentioned in my previous blog post and the Grundlagenstreit was/is another instance of the debate about it?

### the arrow and errors of time

We have memories of the past but not the future. It follows that we use probabilities much more often when we try to predict the future; e.g. a weatherman will tell us that the probability is 35% for rain tomorrow, he will usually not tell us what the probability was for rain yesterday (even if he is uncertain whether it was raining yesterday). The physicists say that this is just an example (and there are many more) of "the arrow of time".

At this point we have a choice.

i) We can declare that this "arrow of time" is a fundamental property of nature and indeed of (human) logic and move on to other topics.

ii) We can try to (and perhaps have to) explain or derive "the arrow of time", because the fundamental laws of physics are invariant under time reversal (with a tiny exception which should not matter for this point) and we have to assume that weathermen and our memories are composed of particles (and strings?) which follow those laws.

As far as I know C.F. v. Weizsaecker was the most prominent physicist in favor of i) and almost all other physicists follow the route ii), using the 2nd law and the assumption of special initial conditions for their explanation/derivation.

But the arrow of time is a tricky topic and not your usual empirical fact. Just try to imagine an experiment which would falsify it. The very existence of physicists debating it already requires this "arrow" or, as a proponent of ii) would argue, it requires a universe in an unlikely state far from equilibrium. This is one reason why there is still a debate about it.

If one is interested in this discussion I recommend e.g. H.D. Zeh's book.

Of course, this is a topic with many opportunities to talk past each other and confuse yourselve and others with circular arguments and errors. In other words, it is an ideal topic for the interwebs.

At this point we have a choice.

i) We can declare that this "arrow of time" is a fundamental property of nature and indeed of (human) logic and move on to other topics.

ii) We can try to (and perhaps have to) explain or derive "the arrow of time", because the fundamental laws of physics are invariant under time reversal (with a tiny exception which should not matter for this point) and we have to assume that weathermen and our memories are composed of particles (and strings?) which follow those laws.

As far as I know C.F. v. Weizsaecker was the most prominent physicist in favor of i) and almost all other physicists follow the route ii), using the 2nd law and the assumption of special initial conditions for their explanation/derivation.

But the arrow of time is a tricky topic and not your usual empirical fact. Just try to imagine an experiment which would falsify it. The very existence of physicists debating it already requires this "arrow" or, as a proponent of ii) would argue, it requires a universe in an unlikely state far from equilibrium. This is one reason why there is still a debate about it.

If one is interested in this discussion I recommend e.g. H.D. Zeh's book.

Of course, this is a topic with many opportunities to talk past each other and confuse yourselve and others with circular arguments and errors. In other words, it is an ideal topic for the interwebs.

Subscribe to:
Posts (Atom)