Mathematics as an Information Process

 ... or "how any brains, human or artificial, can do maths" : including the creation of novel concepts for nouns, verbs, and adjectives in maths.

I started writing this as a comment on Valerio's post, but it overflowed so I made it a repost, then it overflowed, so I had to articleise it.

What the hell is maths?

I think categorically, "mathematics" isn't "a thing that can be solved" as a whole. Mathematical QUESTIONS yes, but there is no end to the questions.

Mathematics ITSELF is a [ software program ], for [ representing ways to reason about phenomenology in general ] - in other words, maths is a programming language, or compiler toolchain - with specific focus on [ phenomenology that is objective ].

( This applies to mathematics as a social activity - in private, any mind can perform subjective quantification upon phenomena it perceives, even though that activity is not directly observable by other minds. Because at this point in history, we still do not have [ social / common access ] to the [ subjective phenomenological experience of humans ], since that memory space is embedded within an ill-understood meat net. )

I think the matter of coming up with new [ mathematics, and science in general ] is really just a generative process of creating ontologies of language where existing ontologies are insufficient. It's just mutate, and cull, loop. What gives rise to the DEMAND for new ontology / concepts, depends a bit more on the field.

PHYSICS :

In physics, ontological DEMAND is a matter of "we have some observational SENSORY data that does a quantifiable pattern, but we don't have a name for the pattern, though we can point at the pattern. Also we don't know how this new pattern relates to old patterns, and we need to do more tests to find out."

MATHEMATICS :

In mathematics, ontological DEMAND is a matter of "basically everything that physicists do, but swap the object of research from SENSORY phenomena in general to [ formal INSENSIBLE verbal concepts, attached to some SENSIBLE notation that we use to represent it ]".

THE DIFFERENCES :

While physicists have to tolerate the limit of "hypotheses are falsifiable, and resistant to falsification under duress," mathematicians tend to need to show that "theses are necessarily true/false".

In the realm of testing the combinatorial play-niceness of known fundamental relations and entities, that's fairly straightforward.

THE COMMONALITIES :

The discovery of new fundamental relations and entities ... is also driven, from the same sensory source that drives such discovery in physics.

Physics and mathematics are both "sensory", it's just that in physics the sensory data is of [ intrinsic ] interest, whereas in maths the sensory data is [ just symbolic of insensible interests ].

So, in the same way machines can be programmed to be creative in hypotheses about [ intrinsically interesting sensory data ], they can do the same for [ sensory data, i.e. language morphemes and syntax, which are extrinsically interesting ].

THE MENTAL MODAL :

After all, in and of itself, [ the latent intuition about how logic, space, and time, work ] is essentially the brain's physical activity, as a [ subconscious memory space ]. Data from there then intrudes, from time to time, upon our also physical [ imaginations, as conscious memory spaces ].

Both memory spaces simply process [ sensory data structures, which we may call qualia when they move into the conscious memory space ].

A practiced thinker knows how to completely map, FROM qualia, TO analogous signals or other data structures which are known to be objectively quantifiable, through the social practice of science.

Of course, I might be venturing a bit too far here, in terms of asserting that meat brains are just sub/conscious physical information processes.

Haha

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Discussion

2026-05-26 :

• the concept of "well if X is in conscious memory, then X must be an instantiation of the data types supported by your conscious memory" is from COPR, as mentioned to Alex : might be worth a wiki; or see the Sophie's World version of Transcendental Idealism / t-analytic / t-aesthetic. Ofc in LeCun's phrasing, this is a "world model".
• I think the insight on maths is : "the only way to handle maths is via sensible symbols, and we are quite sure the LLMs can handle symbols + rules + rules of replacement", so an LLM can look at "any N-dimensional field of raw noise" and formally hypothesise any new set of rules that applies to the field. Given a less noisy field the LLM can run formal or empirical tests against the field, to see if the rules still apply, just for example. This is literally all it takes to invent new science, in a very thin slice lol. And for maths, the bot doesn't even need a noise field for sensory data - it just has to check rule coherence.

2026-05-29:

It is remarkable, after all this time, that mathematics remains as much a veil, to the mode practicioner, in the same way that other conscious macrophenomena are to the mode person. Some people are ontologically committed to the notion of persons, some are committed to the notion of emotions, some are committed to the notion that food is yummy, and travelling is fun. Mathematics likewise is broadly appreciated macroaesthetically.

But if you tear apart the qualia, and isolate the categories of sensational units in every sense modality, then you are left with only a few kinds of lego, so to speak. Mathematics reduces to a sensible language, with insensible referents, and so the treatment of the language can only be tested with regards to the rules employed about the sensible language.

I guess?

2026-06-02:

It's astonishingly stupid, that some professional mathematicians can't understand how it is possible to teach a machine how to discover novel mathematics.

Simply stepping through the history of maths, demonstrates each point of "ontological demand" , where existing linguistic constructs were insufficient for a formal representation of some sensory dataset, thereby leading to the designation on new linguistic constructs as hypothetical frameworks, then affirmed as practically useful via coherence testing.

It is just a special case, of the development of languages in general. Perhaps it is because mathematicians are not often trained as philologists, even though these are both primarily about language use and development. Just as grammar teachers or English professors are not expected to be philologists, as a rule. 

2026-06-07:

Humans spend 150 years making a machine that does weird stuff really fast - because weird stuff is enriching, but humans are slow at it.

Someone figures that the machines could be a little less weird, so makes the weird machine do less weird things, but in a slow way.

Someone then figures out that if you get enough weird machines doing less weird things, in the slow way, you can approximate a human in all its slowness at doing the weird thing. 

Ladies and gentlemen : a neural network doing maths ... instead of outsourcing it to an ALU.