Integers as Compression

post by Chris_Leong · 2020-10-28T07:36:09.192Z · LW · GW · 7 comments

I would like to propose a conception of what integers are. This is something of a misnomer, I don't actually want to claim that there is a one true conception of what integers are, but nonetheless this is a convenient way of describing what this post is about. I actually believe that there's multiple starting points from which we can produce a concept of integers, nonetheless, this appears to me as a particularly beautiful approach and highlights some aspects that other conceptions may not.

Suppose we have the following sequences:

aaaa aaaa

bbbcc bbbcc

abcd abcd

If sequences of this form are fairly common, then the number 2 will be useful for compressing information about these sequences. In particular, we can imagine writing the first sequences as 2{aaaa} instead.

Next let's imagine taking the sequence 2{aaaa} and appending 3{aaaa}. This would give us the sequence 5{aaaa} and leads to a natural notion of addition. 4{2{aaaa}} = 8{aaaa} and so now we have arithmetic as well.

We aren't limited to exact matches either. Consider:

aab aac aax

If the last character is irrelevant, we could write this as 3{aa_}. This is useful for producing a conceptualisation of objects as objects are almost never exactly the same. If we have a photograph, we can imagine utilising a form of lossy compression to represent the data present. For example, we might only care about the fact that there are three people in the photo and not about any of the details. This can further be extended into video and streams of experience. Alternatively, instead of compressing streams of experience, we could imagine compressing some kind of mental model.

One major advantage of talking about compression and sequences instead of objects is that to talk about objects you need to be able to set up a significant amount of metaphysical infrastructure. In contrast, sequences are fairly simple objects. We don't need these sequences to exist in an abstract sense, but only as objects in our mind. We could try defining numbers using abstract objects instead, but these don't naturally provide a notion of addition without something like a notion of space. Sets can provide a useful conception of numbers as well, but the abstract sets in our head are difficult to relate to the concrete sets in the world, while the compression conception links back fairly naturally. But beyond this, I don't believe that we need a unique conception of numbers and that different conceptions can illuminate different aspects of them.

One objection might be that we can't use compression to define numbers because that requires sequences and we need numbers to define sequences formally. This doesn't actually seem like that significant a flaw to me. If someone proposes laws of logic, the only way that we can check that these laws of logic are reasonable is to combine them using some kind of logic or meta-logic. Similarly the only way to check the reliability of our sense (like sight or hearing) is to use sense-data itself to check for consistency. And we can't falsify our assumptions about the world without making assumptions about how the world works in order to interpret our experimental results.

At this point, it is naturally to ask about whether this is useful or just kind of cool. Since as far as I'm concerned this is just a useful conception, not the one true conception of numbers the direct applications of this are somewhat limited. On the other hand, sometimes these conceptions can have indirect applications. For example, the set theory conception of numbers suggests the possibility that "four" mightn't have any meaning outside of something more specific like "four cats", "four dogs" or "four apples".

The compression conception of integers emphasise how numbers appear out of our processing of data, rather than from our experiences of the world. I'm still unsure whether this constitutes evidence for integers existing in the map rather than the territory. But it's interesting to think that we would be able to define the notion of numbers even if there didn't exist an external world.

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