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Science is countable.

Something cannot really be said to be science unless there is a representation which can be communicated between scientists. (If there's an idea in one person's brain, but that person can't transmit it to someone else's brain - you can't call that scientific.) Scientific representations are countable, because every scientific representation can be mapped to a finite list whose elements come from a finite set of symbols.

Mathematics is countable. The set of real numbers is uncountable - but the numbers which can be represented by mathematics is countable. So, there are real numbers which are unrepresentable (and in fact, most of them are unrepresentable).


Um, Wow. In response to Gorgonzola's writeup:

If I had wanted to say that all the science that will be achieved within the (useful) lifetime of this universe is finite, I would have made a node with an appropriate title. If I had wanted to say that all the representations that can be practically made within this entropy-ruled universe is finite, I would have chosen a different node title. I have known these things for quite some time, and I don't think they are really interesting enough to make nodes out of. So I didn't. That's not what I wrote about.

There is one thing I did not make clear in the original post. When I wrote about a "scientist", and a "person"'s "brain", I was using these words in a much more general sense than is normal. I meant to include a hypothetical "person" who is immortal and who has infinite writing supplies which never decay. This person could even have an infinite capacity "brain". The relevant limitations of this person/scientist is that it starts with only a finite amount of information and can only do a finite amount of computation in a finite time. This lack of detail on my part is the only way I can see how Gorgonzola could think that proton decay was in any way related to the content of the writeup, so I apologise for the omission.

I wrote about science in a more abstract sense, not just the science that will be achieved in this limited universe - not even just the science that can be achieved in this universe. The idea is not that narrow, it applies to immortals. Yes, there are no immortals in this universe, yes, the idea is irrelavent in the limited context of this universe. The idea is not about this universe, it's about science.

About confusing "statements about a field derived from a particular meticulous method, with the field itself", the only field of which I speak is science - which is ruled by "meticulous method" (scientific methods, which, by nature of being scientific, can be represented by a finite list of elements from a finite symbol set, - i.e. countable methods). Not to say that there are not other fields, and not to say that science can cover all of any other field... but, science was the only field to which I referred. The idea is not about all fields to which the scientific method may or may not apply, it is about the particular field of science and the parts of other fields which can be embraced by science.

Applying the idea to our limited universe, it's irrelavent. Applying the idea beyond the limits of science, it doesn't hold up. I know that. I was doing neither.

Given dedicated application of some rather specious reasoning, one could come to this conclusion.

Now, there is no such thing as a perpetual motion machine, and the Universe is no exception. Eventually, the entropy in the Universe will expand to the point that no meaningful information can be transmitted. That is, no signal, only noise. Of course, that's a long way away.

But why stop there? This reasoning leads us to the conclusion that science is not only countable, it is finite: Countable infinities cannot be represented any more than uncountable ones can! We don't have enough time to recite all of the positive integers, or represent even one number that requires an infinite decimal (or pick the base of your choice), much less represent all of them!

Ok, so the number of scientific papers that will be published is finite. Better submit now while you can!

But wait: let's imagine an ideal (but practical) calculating machine. The main purpose of the machine is to continually print out positive integers, each one greater than the last. Of course, all sorts of subsystems are attatched to the machine, dedicated to keeping the machine running: To manufacture spare parts for itself after humanity dies out, and to find energy when the sun goes out. Eventually after all of the stars are extinguished, the machine will find ways of exploiting black holes until they, too, all evaporate or are impossibly far away.

The machine will doggedly recite number after number, continuing until it, too, must expire because it can't repair itself fast enough to replace decayed protons.

Eventually the machine will spit out its highest value. Call it N.

Does this mean that N+1 does not exist?

In the end, we have to realize that our reasoning is flawed:
  • Of course, some people may confuse the terms 'countable' and 'infinite'. It's easy to do, but so is learning basic mathematical terminology.

    We can choose to accept the axiom of infinity, or not. Once we accept the existence of even one infinity, however, all the rest inevitably follow.
  • More importantly, we are confusing science, i. e. statements about a field derived from a particular meticulous method, with the field itself. (In other words, we are confusing the doctor with the patient.)

    In regard to mathematics, the number of mathematical statements we can practically make is unfortunately finite. If you concatenate all of the mathematical literature likely to be produced from now until the end of time, and then represent it as a number, you will get a very, very large, but nonetheless finite, number. Call it L. Does L+1 exist?. I can't think of a mathematician willing to give up the notion of the natural numbers being closed for the operation of addition.

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