Got it: https://en.wikipedia.org/wiki/Information_mapping
Polytopos
[Question] Why hasn’t the technology of Knowledge Representation (i.e., semantic networks, concept graphs, ontology engineering) been applied to create tools to help human thinkers?
Interesting, can you give some examples to illustrate how causal/Bayes nets are used to aid reasoning / discovery?
I see merit in the idea that semantic networks may focus too much on the structure of language, and not enough on the structure of the underlying domain being modelled. As active thinkers, we are looking to build an understanding of the domain, not an understanding of how we talked about that domain.
Issues of language use, such as avoiding ambiguity, could sometimes be useful especially in more abstract argumentation, but more important is being able to track all of the relationships among the domain specific entities and organizing lines of evidence.
Mathematical Intuitionism and the Flow of Time
Thanks for your comment. My replies are below.
“so Gisin’s musings… are guaranteed to be not a step in any progress of the understanding of physics.”
What is your epistemic justification for asserting such a guarantee of failure? Of course, any new speculative idea in theoretical physics is far from likely to be adopted as part of the core theory, but you are making a much stronger claim by saying that it will not even be “a step in any progress of the understanding of physics”. Even ideas that are eventually rejected as false, are often useful for developing understanding. Gisin’s papers ask physicists to consider their unexamined assumptions about the nature of math itself, which seems at least like a fruitful path of inquiry, even if it won’t necessarily lead to any major breakthroughs.
“mathematical proofs are as much observations as anything else. Just because they happen in one’s head or with a pencil on paper, they are still observations.”
This reminds me of John Locke’s view that mathematical truths come from observation of internal states. That is an interesting perspective, but I’m not sure it an hold up to scrutiny. The biggest issue with it seems to be that in order to evaluate the evidence provided by empirical observations we must have a rational framework which includes logic and math. If logic and math themselves were simply observational, then we have no framework for evaluating the evidence provided by those observations. Perhaps you can give an alternative account of how we evaluate evidence without pre-supposing a rational framework.
“The difficulty of calculating a far-away digit in the decimal expansion of pi has nothing to do with pi itself: you can perfectly well define it as the ratio of circumference to diameter, or as a limit of some series”
I agree with this statement. I think though it misses the point I was elaborating about Brouwer’s concept of choice sequences. The issue isn’t that we can’t define a sequence that is equivalent to the infinite expansion of pi, I think it is rather that for any real quantity we an never be certain that it will continue to obey the lawlike expansion into the future. So the issue isn’t the “difficulty of calculating a far-away digit” the issue is that no matter how many digits we observe following the law like pattern, the future digits may still deviate from that pattern. No matter how many digits of pi a real number contains, the next digit might suddenly be something other than pi (in which case we would say retrospectively that the real number was never equal to pi in the first place). This is actually what we observe, if we are to say measure the ratio of a jar lid’s diameter to it’s circumference. The first few digits will match pi, but then as we to smaller scales it will deviate.
″...the idea that Einstein’s equations are somehow unique in terms of being timeless is utterly false”
I made no claim that they are unique in this regard.
David Spivak offers an account of Categories as database schemas with path equivalencies that is similar to the account you’ve given here in his book Category Theory for the Sciences. He still presents the traditional definitions, giving examples mainly from the category of sets and functions. I also didn’t find his presentation of database schema definition especially easy to understand, but it is very useful when you realize that a functor is a systematic migration of data between schemas.
It seems odd to equate rationality with probabilistic reasoning. Philosophers have always distinguished between demonstrative (i.e., mathematical) reasoning and probabilistic (i.e., empirical) reasoning. To say that rationality is constituted only by the latter form reasoning is very odd, especially considering that it is only though demonstrative knowledge that we can even formulate such things as Bayesian mathematics.
Category theory is a meta-theory of demonstrative knowledge. It helps us understand how concepts relate to each other in a rigorous way. This helps with the theory side of science rather than the observation side of science (although applied category theories are working to build unified formalisms for experiments-as-events and theories).
I think it is accurate to say that, outside of computer science, applied category theory is a very young field (maybe 10-20 years old). It is not surprising that there haven’t been major breakthroughs yet. Historically fruitful applications of discoveries in pure math often take decades or even centuries to develop. The wave equation was discovered in the 1750s in a pure math context, but it wasn’t until the 1860s that Maxwell used it to develop a theory of electromagnetism. Of course, this is not in itself an argument that CT will produce applied breakthroughs. However, we can draw a kind of meta-historical generalization that mathematical theories which are central/profound to pure mathematicians often turn out to be useful in describing the world (Ian Stewart sketches this argument in his Concepts of Modern Mathematics pp 6-7).
CT is one of the key ideas in 20th century algebra/topology/logic which has allowed huge innovation in modern mathematics. What I find interesting in particular about CT is how it allows problems to be translated between universes of discourse. I think a lot of its promise in science may be in a similar vein. Imagine if scientists across different scientific disciplines had a way to use the theoretical insights of other disciplines to attack their problems. We already see this when say economists borrow equations from physics, but CT could enable a more systematic sharing of theoretical apparatus across scientific domains.
I disagree with the idea that one doesn’t have intuitions about generalization if one hasn’t studied mathematics. One things that I find so interesting about CT is that it is so general it applies as much to everyday common sense concepts as it does to mathematical ones. David Spivak’s ontology logs are a great illustration of this.
I do agree that there isn’t a really good beginners book that covers category theory in a general way. But there are some amazing YouTube lectures. I got started on CT with this series, Category Theory for Beginners. The videos are quite long, but the lecturer does an amazing job explaining all the difficult concepts with lots of great visual diagrams. What is great about this series is that despite the “beginners” in the title he actually covers many more advanced topics such as adjunction, Yoneda’s lemma, and topos theory in a way that doesn’t presuppose prior mathematical knowledge.
In terms of books, Conceptual Mathematics really helped me with the basics of sets and functions, although it doesn’t get into the more abstract stuff very much. Finally, Category Theory for Programmers is quite accessible if you have any background in computer programming.
I was excited by the initial direction of the article, but somewhat disappointed with how it unfolded.
In terms of Leibniz’s hope for a universal formal language we may be closer to that. The new book Modal Homotopy Type Theory (2020 by David Corfield) argues that much of the disappointment with formal languages among philosophers and linguists stems from the fact that through the 20th century most attempts to formalize natural language did so with first-order predicate logic or other logics that lacked dependent types. Yet, dependent types are natural in both mathematical discourse and ordinary language.
Martin-Lof developed the theory of dependent types in the 1970s and now Homotopy Type Theory has been developed on top of that to serve as a computation-friendly foundation for mathematics. Corfield argues that such type theories offer new hope for the possibility of formalizing the semantic structure of natural language.
Of course, this hasn’t been accomplished yet, but it’s exciting to think that Leibniz’s dream may be realized in our century.
Interesting. This might be somewhat off topic, but I’m curious how would such an Bayesian analysis of mathematical knowledge explain the fact that it is provable that any number of randomly selected real numbers are non-computable with a probability 1, yet this is not equivalent to a proof that all real numbers are non-computable. The real numbers 1, 1.4, square root 2, pi, etc are all computable numbers, although the probability of such numbers occurring in an empirical sample of the domain is zero.
I don’t know enough math to understand your response. However, from the bits I can understand, it seems leave open the epistemic issue of needing an account of demostrative knowledge that is not dependent on Bayesian probability.
I second this book recommendation. I just finished reading it and it is well written and well argued. Bregman explicitly contrasts Hobbes’ pessimistic view of human nature with Rousseau’s positive view. According to the most recent evidence Rousseau was correct.
His evolutionary argument is that social learning was the overwhelming fitness inducing ability that drove human evolution. As a result we evolved for friendliness and cooperation as a byproduct of selection for social learning.
[Question] How can labour productivity growth be an indicator of automation?
I find reading this post and the ensuing discussion quite interesting because I studied academic philosophy (both analytic and continental) for about 12 years at university. Then I changed course and moved into programming and math, and developed a strong interest thinking about AI safety.
I find this debate a bit strange. Academic philosophy has its problems, but it’s also a massive treasure trove of interesting ideas and rigorous arguments. I can understand the feeling of not wanting to get bogged down in the endless minutia of academic philosophizing in order to be able to say anything interesting about AI. On the other hand, I don’t quite agree that we should just re-invent the wheel completely and then look to the literature to find “philosophical nearest neighbor”. Imagine suggesting we do that with math. “Who cares about what all these mathematicians have written, just invent your own mathematical concepts from scratch and then look to find the nearest neighbor in the mathematical literature.” You could do that, but you’d be wasting a huge amount of time and energy re-discovering things that are already well understood in the appropriate field of study. I routinely find myself reading pseudo-philosophical debates among science/engineering types and thinking to myself, I wish they had read philosopher X on that topic so that their thinking would be clearer.
It seems that here on LW many people have a definition of “rationalist” that amounts to endorsing a specific set of philosophical positions or meta-theories (e.g., naturalism, Bayesianism, logical empiricism, reductionism, etc). In contrast, I think that the study of philosophy shows another way of understanding what it is to be a rational inquirer. It involves a sensitivity to reason and argument, a willingness to question one’s cherished assumptions, a willingness to be generous with one’s intellectual interlocutors. In other words, being rational means following a set of tacit norms for inquiry and dialogue rather than holding a specific set of beliefs or theories.In this sense of reason does not involve a commitment to any specific meta-theory. Plato’s theory of the forms, however implausible it seems to us today, is just as much an expression of rationalism in the philosophical sense. It was a good-faith effort to try to make sense of reality according to best arguments and evidence of his day. For me, the greatest value of studying philosophy is that it teaches rational inquiry as a way of life. It shows us that all these different weird theories can be compatible with a shared commitment to reason as the path to truth.
Unfortunately, this shared commitment does break down in some places in the 19th and 20th centuries. With certain continental “philosophers” like Nietzsche, Derrida and Foucault their writing undermines the commitment to rational inquiry itself, and ends up being a lot of posturing and rhetoric. However, even on the continental side there are some philosophers who are committed to rational inquiry (my favourite being Merleau-Ponty who pioneered ideas of grounded intelligence that inspired certain approaches in RL research today).
I think it’s also worth noting that Nick Bostrum who helped found the field of AI safety is a straight-up Oxford trained analytic philosopher. In my Master’s program, I attended a talk he gave on Utilitarianism at Oxford back in 2007 before he was well known for AI related stuff.
Another philosopher who I think should get more attention in the AI-alignment discussion is Harry Frankfurt. He wrote brilliantly on value-alignment problem for humans (i.e., how do we ourselves align conflicting desires, values, interests, etc.).
Agreed. Open AI did a study on the trends of algorithm efficiency. They found a 44x improvement in training efficiency on ImageNet over 7 years.
I find it hard to believe your prediction that this breakthrough will be insignificant given what I’ve read in other reputable sources. I give a pretty high initial credence to the scientific claims of publications like Nature which had this to say in their article on Alphafold2:
The ability to accurately predict protein structures from their amino-acid sequence would be a huge boon to life sciences and medicine. It would vastly accelerate efforts to understand the building blocks of cells and enable quicker and more advanced drug discovery.
There is a wonderful scene in the new Pixar film Soul where they show a “lost soul” who turns out to be a hedge fund trader who just keeps saying, “gotta make the trade”. Your description of your high income clients reminded me of that.
[Question] What do you think should be included in a series about conceptual media?
The trouble is that these antibodies are not logical. On the contrary; these antibodies are often highly illogical. They are the blind spots that let us live with a dangerous meme without being impelled to action by it.
That is a brilliant point. I also loved you description of the Buddhist monk taking questions from a Western audience. The image of incompatible knowledge blocks is a great one, that actually makes a lot of sense of how various ideologically conditioned people are able to functionally operate.
The example that comes up for me is animal suffering. I believe that torturing animals in factory farms will one day be regarded as a moral evil on bar with war, slavery, etc. I while I refrain from meat, I have a blindspot for eggs and milk, and still a bigger blind spot for other’s eating meat. If I didn’t have the latter blindspot, I wouldn’t be able to function in society. I don’t go around consciously thinking of factory carnivores as moral monsters. Maybe if I were more rationally driven I would think this way, and that might be a very bad thing.
Maybe the true judo move is to learn how to include the practical rationality of when to compartmentalize in the rational calculus. Of course, we might not have such fine grained control over these unconscious aspects of our cognition.
Hi Said. I’m new here, would you mind explaining what a sidebar is, maybe providing a link or instructions to find said sidebar? Thanks.