On: Don’t Kill Math

Link: Don’t kill math

A case for analytic method of learning vs a more visual one. There are obvious advantages to learning math the simulation/visualization way and the symbolic way. There are two points that come to my mind after reading this:

• Bret Victor’s “new“ approach isn’t the first one that tries to deal with mathematics in a post-computing era. While many mathematicians utilise modern computers in their daily research, the original method for learning mathematics perhaps hasn’t changed in decades (this is, however, not from first-hand experience, but by reading articles from other mathematicians). Data visualization, Python or R have become ubiquitous in statistics and machine learning. Since I am not a physicist I can’t be one hundred per cent sure, but even Physics now heavily depends on computer simulations and data analysis.

• But “pure“ mathematics seems to have eluded computer utilization up until recently. When it comes to higher level, abstract math, I can think of only two ways to move away from the symbolic approach: theorem provers and LLMs. The latter is an immature and extremely infant technology, hence we must wait to see it’s converging destination. However, the former is an interesting topic. To be precise:
  Why not use theorem provers to even teach mathematics, and build intuition ?

What I am suggesting is, in my opinion, the modernization of mathematics. The traditional methods of learning mathematics can be replaced by modern strongly and extremely expressively typed programming languages, which have multiple benefits.

• Instead of relying on arbitrary problem-solving and back-of-the-chapter questions to build strong intuition, we are literally forced to build intuition by explicitly writing various concepts in a programming language.

• Proofs are immediately checked by the system, providing faster feedbacks (as opposed to having another person check ones proofs).

• Any arbitrary definition can be shared, without past material or context (due to each mathematician having their own compiler and libraries). This could give math the same reach and democratizing effect as computer science.

Perhaps this is already being followed, I am not sure. But these advantages are there, and force one to remove any and all ambiguities about mathematics (especially pure and higher-level abstract) by writing concrete and typed definitions.

That being said, programming languages (even ones like Idris, Agda, Lean or Coq) have a long way to go before being expressive enough to encode some mathematical theories. Thinking in terms of types may also somehow constrain ones thinking in a particular direction, and bias certain results while obscuring others. This would be made clear if there is a mass adoption of learning math purely through a typed programming language.

On: Follow The Converging Lines of Evidence

Link: Follow The converging lines of evidence

Good arguments for the existence of God. Although the discoverability of Universe being astonishing, and mathematics being this accurate seeming like a miracle, seems quite far fetched. It is indeed astonishing how accurate mathematics is, but this seems to be a case of survivorship bias. We didn’t get the accuracy without trying and failing over two thousand years, hundreds of thousands of man-hours being spend on trying to discover the truth. Even then we only got so far, with a fundamental understanding of reality still being elusive. Not to mention other things, which where previously hidden from us (like our own biology, consciousness, etc…) coming into the lime light only recently.

I am aware that the above argument fails to take into account why humans are intelligent in the first place. Intelligence seems more of a consequence of evolution then a phenomenon with a “purpose“ or goal. However this isn’t a logical counter-point, but rather a belief, but a belief that is still grounded in something that is falsifiable.

As for the rest of the argument, which is that the convergence of various lines of evidence all point to a single explanation, and how various arguments against God are all inherently the same point expressed in different ways (which kind of invalidates them), there’s this comment and this article, which provide counter arguments, which I agree with. However, I can be biased here.

On: Emergence

Links :

Relative explanatory autonomy

Emergence

One aspect of emergence that the above mentioned essays point out is how a “higher“ level phenomena, one that emerges, is more than the sum of it’s parts: it cannot be reduced to a simple derivation of it’s components.

This non-linearity, in my opinion, comes from another property, which is computational complexity, or rather, computational irreducibility. This property could potentially act as a barrier, one enforced by the laws of physics (this means an absence of clever heuristics as well, not just a failure of brute force) that stops us from reducing everything to it’s most atomic building block, and which could force us to find different explanatory principles, on the other side of the said barrier.

Some examples could be neurons → intelligence, particle physics → biology/chemistry, solving a chess game by brute force or, and this is the least likely, going from weights → full explainability of LLM behaviour.

On: Eternalist Systems

Link: Eternalist Systems

This is a post-rationalist flavoured blog (since my understanding of post-rationalism is very limited, this is only an opinion), or more accurately a part of the book.

To put this blog in one sentence:

It all comes down to faith

No matter the belief system, if one subscribes to a particular belief, one requires faith to do so. No amount of “logic“, “rationalism“ or holy scriptures can convince a person to find meaning in their lives, unless it’s backed by faith.

One other curious concept the author mentions is “Nebulosity“, which in some sense seems like a semantic derivation of entropy ? The author tries to pin down the meaning of that word to:

the insubstantial, amorphous, non-separable, transient, ambiguous nature of meaningness

Another word being meaningness. This word, to my understanding, is the “middle point“ between two thoughts: meaninglessness (nihilism) and meaningfulness (existentialism). This seems very close to another ideology from this tweet:

Basically either side has it’s downfall (since everything is nebulous), and we would be much better if we stayed somewhere in the middle (which I think could be optimistic nihilism, which is also very similar to the humour mentioned in the tweet).

On: Extremes

Links :

Hubris and Control

Here the author, Duncan Sabien practically points towards another obvious-but-overlooked fact (as he does usually in his essays), which is that a lot of the things/rules/methods which we follow have not been well thought out, in-context with our evolutionary purpose.

Our body’s signals are a very unreliable way to go about deciding the methodologies for almost all things modern: since the signals were wired tens or hundreds of thousands of years ago, they have not well-adapted to our modern way of life, and can even prove to be detrimental to our health (one of the prime examples is auto-immune disorders). The analogy he uses is that of over-optimization ad infinitum: where we give in to our desires because “if feels right“, not realizing our desires/feelings have not accounted for modern technologies, economies and social networks.

He mentions this as us taking things to the extreme, and I’d like to connect it to one of my pieces, where I point out that any belief, taken to the extreme, is most probably false. This could be extended, wherein we can say, extremes are usually bad/detrimental.

The way I see it, Mr. Sabein’s point is a specific instantiation of a more general principle which I have mentioned (which coincidentally was a fundamental proof in probability). In general, extremes are bad/detrimental/less probable.

On: Truth or Dare

Link: Truth or Dare

This turned out to be one of those long yet engaging blogs I found myself reading from gmail. This blog (and the one referenced multiple times by the author) talk about how different people, given the same environment, act and react in completely different ways.

The post that the author references, by Scott Alexander, assumes that the reason for people finding themselves in certain situations, with certain kinds of people is due to their own personality traits and the environment they’ve been exposed to. The author continues on this point, mentioning how different people may experience the same thing, yet come to completely difference conclusions, since they went in with completely different priors.

It is very evident that this would be the case, especially if one is fortunate enough to interact with people from different walks of life. I have met people with completely different views then mine, which seem to be in direct conflict with my observations. Apart from views, I see people also seem to find themselves in certain predicaments which, from my perspective, seemed completely avoidable, yet the person embraces it as if it were a norm (“such is life“ or “life is hard, get used to it“). Such acts lead me to believe that people not only have different experiences, but also learn behaviour in response to those experiences, which in-turn expose them to similar experiences … and such is life.

This seems to comply in almost all aspects of life. “Birds of a feather flock together“. A few examples are: someone who’s worried vs someone who isn’t, to get a job. The latter ends with a job, while the former struggles. (This would remind one of something like “manifestation“, but only this time, it’s internal and unconscious).