Can we adequately explain complex phenomenon through a single number?

Question

Think of the most complex number you can. Say a surd like pi. What does it explain, exactly? How complex is the ratio of the circumfrence to the diameter of a circle?

I think numbers can describe relationships, which is why you need formula of multiple variables. A single number doesn't describe a relationship. Only because the problem of the relationship of the diameter to the circumfrence of a circle is so infamous do we even know what pi is. Otherwise, pi itself doesn't describe a thing.

Also , You can put all the details of a problem into a single number (e.g 2424, 362, represent it as 24240362) where these correspond to different details of the phenomenon although a single number won't represent all the data you need to represent the problem. That's analoguous to using a number as the output of a coding algorithm. You still need all the ground rules of how to take the separate numbers apart and apply them together, and that info is outside the aggregated number. So the aggregated number itself is just a number. You haven't put all the details into it, nor would it be possible to do so.

You can say you will always need a common language for this since language is a protocol for how to extra meaning from this raw data. If both parties have the same rules then it is possible , i.e they have a common protocol (like having a common language). However it's analogous to an encryption code (a common language). This means that you can't put "all the details of a problem" into a single number, if you can't also encode the rules of use into it. You'd be using the single number as a field of data, again, much like an encrypted message (which can often be represented as one large number).

My motivations for typing these thoughts out , is how skeptical I am about how such a complicated function such as intelligence , personality or creativity can be described through a single number.

Answer 1

You're asking the wrong question.

In order to get there, though, let's step through a few of your claims.

A single number doesn't describe a relationship.

Well, that's trivially true-- but usually a number is used in a given context. For example, if we say "78 percent", we are describing a relationship-- that between 78 and 100.

Also , You can put all the details of a problem into a single number (e.g 2424, 362, represent it as 24240362) where these correspond to different details of the phenomenon although a single number won't represent all the data you need to represent the problem. That's analoguous to using a number as the output of a coding algorithm. You still need all the ground rules of how to take the separate numbers apart and apply them together, and that info is outside the aggregated number. So the aggregated number itself is just a number. You haven't put all the details into it, nor would it be possible to do so.

Nor would you have any need to do so. No one using a single-number measurement is expecting that number to be meaningful outside of context. Nor, for that matter, could any written text contain the instructions for how to read the language it is written in to be used by someone not already able to read that language.

My motivations for typing these thoughts out , is how skeptical I am about how such a complicated function such as intelligence , personality or creativity can be described through a single number.

Any function can be described as a single number, if that function is viewed as a measurement. The question you should be asking, instead of taking this blind alley regarding numbers, etc., is whether "intelligence" or "creativity" or "personality" describe measurable functions (like "length" or "weight", etc.)

If intelligence (for example) is measurable, we can come up with an appropriate scale to represent that measurement. If not, the exercise is doomed to failure.

Note that this has absolutely nothing to do with the (information-theoretic) question of "how much information can be contained in a single number?"

Answer 2

This question, it seems to me, is conflating two ideas. One is generally known as the Library of Babel, after Borges' famous story of the same name, about a library containing all possible combinations of letters. If such a library could exist, then it follows from the premise that somewhere in the library is a book with ultimate wisdom in it --just by random chance.

Theoretically, an irrational number like pi is a Library of Babel, containing all possible combinations of numbers (which can be translated to letters with a simple substitution cypher, such as "A = 01", "B = 02"). Like the oeuvre of the infinite monkeys with the typewriters, this library will supposedly contain all works of literature somewhere within it, but in a wholly un-indexable manner. With that said, it is entirely possible for an infinitely long, non repeating number to not include all possible numerical combinations of all lengths --I'm not aware of whether it has been proven or not that pi does in fact contain all possible combinations (and without that, the entire concept collapses).

The other idea is superficially similar, but actually quite different. There is a cryptographic method that can theoretically reproduce a message of any length with a single mark. The concept is that you translate the message into a decimal less than one, using the same kind of substitution cypher as described above. Then mark a rod or some other object at the point corresponding to that exact ratio. The desired target of your communication receives the object, and measures the mark to reconstruct that number. Of course, in practice, the method is limited by the level of precision of the measurement, so it really only works for extremely short messages. In this second case, a single number can convey a large amount of information, but it is only the information that the message sender has deliberately encoded in this manner.

Answer 3

This is a subtle question. First, it depends on what you mean by a number:

Traditionally one would think of a number in the form you've suggested.An integer or a rational or a real. I've enumerated them in complexity with the last having infinite complexity. One could - in theory, but not in practise - via encoding transcribe all the contents of every book that has been printed in every country into one real number. It's a great idea for electronic communication - but somehow this doesn't feel very satisfying.

In more contemporary idiom, a number can be a more subtil beast - a whole number system such as the reals, or a group; or perhaps the entire category of groups; or by a modern updating of Descartes magic trick a geometric figure or even an entire geometry. This offers a more satisfying possibility of representation. For example certain complex phenomena in mathematics were described by integers and then it was understood that they were more satisfyingly described by geometric shapes or by number systems.

If one then subscribes to materialism, and in particular to physicalism then the entire universe can be putatively described by some geometry or geometric shape which can also be understood as a number.

This makes everything nice and tidy - if one subscribes to physicalism; if one doesn't (as I do) then its a difficult and probably impossible question - intelligence is not a measurable in the mathematical or physical sense.

But even here, that is in Physicalism, an underhand sleight-of-hand trick was executed. A sin of omission: omitting the observer.

For example, a number is rendered meaningless without a consciousness to understand it. A number may inhabit some ethereal platonic realm, but in the world it has to be represented; and that representation is only some arbitrary figure without some consciousness to give it meaning as number.