141–150 of 211 posts
When I used to teach a class called “Mathematics for Liberal Arts Majors,” the first class began with having students “draw six.¹” I refused to give more explicit instructions. It was fascinating to see the results that came out of this, along with the ideas about numbers that it sparked. 1. I have to confess that the choice of six was not arbitrary.
While I am not a big fan of Wolfram’s physics (I think it’s a bit backwards that he took something that he knows very well and somehow finds that it’s how the universe works), I kind of like the idea that space and time emerge from something more computational.
I think that’s what the article should have been focused on...
I think numbers are wonderful and cool abstract thingies. We grow up being taught them (in my case Arabic numerals in base 10) and they become such part of our being that we start to think that we know what “5” is. Then I was introduced to Roman Numerals. Didn’t like those much. That one taught me that 4 is more related to 5 than it is 3. Then some of us learn a base 2 number system or base 16 number system and for a…
When I used to teach a class called “Mathematics for Liberal Arts Majors,” the first class began with having students “draw six.¹” I refused to give more explicit instructions. It was fascinating to see the results that came out of this, along with the ideas about numbers that it sparked. 1. I have to confess that the choice of six was not arbitrary.
Why six?
While the post covers quite a bit of ground, it feels (to me) like it conflates knowledge representation, language, biological systems (i.e., the messiness of implementation), computability, and realism. Regarding numbers in particular, there are a practically uncountably infinite number of mathematical truths that apply equally to numbers or to other abstract (non-numerical) mathematical ideas. I would rather see de…
Because of the limitation of language there's only a countable number of mathematical truths that can be proven or written down. So for all practical purposes there's countably many.
That does not mean that all truths can be written down or enumerated, but strictly speaking it is not sufficient to conclude that due to the fact that the set of all proofs are countable, that the set of all truths entailed by the proofs must also be countable.
None of this should be taken to violate Godel's incompleteness theorems.
Finally, it's worth mentioning that not all formal systems are limited to finite proofs. There are formal systems where theorems as well as proofs can be countably infinite in length and where there are uncountably many proofs. These systems, known as infinitary logic, are often reduceable to second order logic and hence are incomplete.
If Mathematics is a language to describe reality, maybe numbers are not its whole alphabet. Inherent complexity, things that are not discrete, and emergent properties may not be described adequately with numbers, and maybe a different alphabet or even language is needed. Numbers may be (or not, it may depend on our biology) a good initial concept, but maybe something else may be developed, something more "correct" to…
https://youtu.be/GAcUZ3my6E0?t=480
Actually such computer would look like a universe, moving particles around in a continous space.
I guess one of the most fundamental difference between Wolfram’s model for fundamental physics and traditional physics is that Wolfram’s doesn’t have the concept of measure or of continuum at the fundamental level. Space and time, according to Wolfram’s model of the universe, are emerging properties of ‘the network’. Without such things as space and measures, there is no numbers in the fundamental “equations” that dr…
That's in a sense how every model works. All theoretical models of the world are human inventions that aid in making sense of the world in terms familiar to us and nothing more. If you get good results from thinking the world is made of atoms, then the world is made out of atoms. When someone comes along and explains the same thing with strings, then the world is made out of strings. Models are just 'manners of speaking'. It might very well be that you have a dozen entirely different, but equally accurate fundamental ways to talk about a thing.
I like that he keeps it open-minded, but I've thought a lot about this and to me the case for numbers being inevitable is decisively yes. Every adaptable system evolves to adapt to a changing environment by detecting "modes" (categories) and adapting to each mode. Then we start noticing categories come in instances. There's a tree, there are more trees. So now it's useful to count them... Then it's useful to have fra…
I really like this part of the explanation, although not all adaptive systems are intelligent. E.g if I add a grain of salt on a heap of salt and it suddenly collapses, that is an adaptive system but not as a result of intelligent modeling of how it should behave.
> Not because numbers are fundamental to the universe. Rather they're fundamental aspect of intelligently adapting to (i.e. thinking about and making predictions about) the universe.
Here the problem is assuming intelligent adaptivity and the agent that has it is something separate from the universe. Being able to have a model of the universe inherently requires internally resembling the structural functional organization of the universe, a mutual conformity if you will, and therefore if numbers, counting, modes, categories etc are fundamentally useful constructs, I think they also at least resemble a fundamental part of the universe.
How many people are we? How many berries do we need to collect for each.
Later: How many seeds do we need to sow. How big a foundation is needed for this building… How many suns until the weather gets warmer again?
I suppose you could do without if you lived in a cave with endless food readily available outside.