Web of C-lief: conjectures vs. model assumptions vs. scientific beliefs

A sketch of the theoretical computer science Web of C-lief weaved by the non-contradiction spider.

In his 1951 paper on the “Two Dogmas of Empiricism”, W.V.O Quine introduced the Web of Belief as a metaphor for his holistic epistemology of scientific knowledge. With this metaphor, Quine aimed to give an alternative to the reductive atomising epistemology of the logical empiricists. For Quine, no “fact” is an island and no experiment can be focused in to resole just one hypothesis. Instead, each of our beliefs forms part of an interconnected web and when a new belief conflicts with an existing one then this is a signal for us to refine some belief. But this signal does not unambiguously single out a specific belief that we should refine. Just a set of beliefs that are incompatible with out new one, or that if refined could bring our belief system back into coherence. We then use alternative mechanisms like simplicity or minimality (or some aesthetic consideration) to choose which belief to update. Usually, we are more willing to give up beliefs that are peripheral to the web — that are connected to or change fewer other beliefs — than the beliefs that are central to our web.

In this post, I want to play with Quine’s web of belief metaphor in the context of science. This will force us to restrict it to specific domains instead of the grand theory that Quine intended. From this, I can then adapt the metaphor from belief in science to c-liefs in mathematics. This will let me discuss how complexity class seperation conjectures are structured in theoretical computer science and why this is fundamentally different from model assumptions in natural science.

So let’s start with a return to the relevant philosophy.

Process over state: Math is about proofs, not theorems.

A couple of days ago, Maylin and I went to pick blackberries along some trails near our house. We spent a number of hours doing it and eventually I turned all those berries into one half-litre jar of jam.

On the way to the blackberry trails, we passed a perfectly fine Waitrose — a supermarket that sells (among countless other things) jam. A supermarket I had to go to later anyways to get jamming sugar. Why didn’t we just buy the blackberries or the jam itself? It wasn’t a matter of money: several hours of our time picking berries and cooking them cost much more than a half-litre of jam, even from Waitrose.

I think that we spent time picking the berries and making the jam for the same reason that mathematicians prove theorems.

Imagine that you had a machine where you put in a statement and it replied with perfect accuracy if that statement was true or false (or maybe ill-posed). Would mathematicians welcome such a machine? It seems that Hilbert and the other formalists at the start of the 20th century certainly did. They wanted a process that could resolve any mathematical statement.

Such a hypothetical machine would be a Waitrose for theorems.

But is math just about establishing the truth of mathematical statements? More importantly, is the math that is written for other mathematicians just about establishing the truth of mathematical statements?

I don’t think so.

Math is about ideas. About techniques for thinking and proving things. Not just about the outcome of those techniques.

This is true of much of science and philosophy, as well. So although I will focus this post on the importance of process over state/outcome in pure math, I think it can also be read from the perspective of process over state in science or philosophy more broadly.

Reductionism: to computer science from philosophy

A biologist and a mathematician walk together into their joint office to find the rubbish bin on top of the desk and on fire. The biologist rushes out, grabs a fire extinguisher, puts out the blaze, returns the bin to the floor and they both start their workday.

The next day, the same pair return to their office to find the rubbish bin in its correct place on the floor but again on fire. This time the mathematician springs to action. She takes the burning bin, puts it on the table, and starts her workday.

The biologist is confused.

Mathematician: “don’t worry, I’ve reduced the problem to a previously solved case.”

What’s the moral of the story? Clearly, it’s that reductionism is “[o]ne of the most used and abused terms in the philosophical lexicon.” At least it is abused enough for this sentiment to make the opening line of Ruse’s (2005) entry in the Oxford Companion to Philosophy.

All of this was not apparent to me.

I underestimated the extent of disagreement about the meaning of reductionism among people who are saying serious things. A disagreement that goes deeper than the opening joke or the distinction between ontological, epistemological, methodological, and theoretical reductionism. Given how much I’ve written about the relationship between reductive and effective theories, it seems important for me to sort out how people read ‘reductive’.

Let me paint the difference that I want to discuss in the broadest stroke with reference to the mind-body problem. Both of the examples I use are purely illustrative and I do not aim to endorse either. There is one sense in which reductionism uses reduce in the same way as ‘reduce, reuse, and recycle’: i.e. reduce = use less, eliminate. It is in this way that behaviourism is a reductive account of the mind, since it (aspires to) eliminate the need to refer to hidden mental, rather than just behavioural, states. There is a second sense in which reductionism uses reducere, or literally from Latin: to bring back. It is in this way that the mind can be reduced to the brain; i.e. discussions of the mind can be brought back to discussions of the brain, and the mind can be taken as fully dependent on the brain. I’ll expand more on this sense throughout the post.

In practice, the two senses above are often conflated and intertwined. For example, instead of saying that the mind is fully dependent on the brain, people will often say that the mind is nothing but the brain, or nothing over and above the brain. When doing this, they’re doing at least two different things. First, they’re claiming to have eliminated something. And second, conflating reduce and reducere. This observation of conflation is similar to my claim that Galileo conflated idealization and abstraction in his book-keeping analogy.

And just like with my distinction between idealization and abstraction, to avoid confusion, the two senses of reductionism should be kept conceptually separate. As before, I’ll make this clear by looking at how theoretical computer science handles reductions. A study in algorithmic philosophy!

In my typical arrogance, I will rename the reduce-concept as eliminativism. And based on its agreement with theoretical computer science, I will keep the reducere-concept as reductionism.

Plato and the working mathematician on Truth and discourse

Plato’s writing and philosophy are widely studied in colleges, and often turned to as founding texts of western philosophy. But if we went out looking for people that embraced the philosophy — if we went out looking for actual Platonist — then I think we would come up empty-handed. Or maybe not?

A tempting counter-example is the mathematician.

It certainly seems that to do mathematics, it helps to imagine the objects that you’re studying as inherently real but in a realm that is separate from your desk, chair and laptop. I am certainly susceptible to this thinking. Some mathematicians might even claim that they are mathematical platonists. But there is sometimes reasons to doubt the seriousness of this claim. As Reuben Hersh wrote in Some Proposals for Reviving the Philosophy of Mathematics:

the typical “working mathematician” is a Platonist on weekdays and a formalist on Sundays. That is, when he is doing mathematics, he is convinced that he is dealing with an objective reality whose properties he is attempting to determine. But then, when challenged to give a philosophical account of this reality, he finds it easiest to pretend that he does not believe in it after all.

What explains this discrepency? Is mathematical platonism — or a general vague idealism about mathematical objects — compatible with the actual philosophy attributed to Plato? This is the jist of a question that Conifold asked on the Philosophy StackExchange almost 4 years ago.

In this post, I want to revisit and share my answer. This well let us contrast mathematical platonism with a standard reading of Plato’s thought. After, I’ll take some helpful lessons from postmodernism and consider an alternative reading of Plato. Hopefully this PoMo Plato can suggest some fun thoughts on the old debate on discovery vs invention in mathematics, and better flesh out my Kantian position on the Church-Turing thesis.

Bourbaki vs the Russian method as a lens on heuristic models

There are many approaches to teaching higher maths, but two popular ones, that are often held in contrast to each other, are the Bourbaki and Russian methods. The Bourbaki method is named after a fictional mathematician — a nom-de-plume used by a group of mostly French mathematicians in the middle of the 20th century — Nicholas Bourbaki, who is responsible for an extremely abstract and axiomatic treatment of much of modern mathematics in his encyclopedic work Éléments de mathématique. As a pedagogical method, it is very formalist and consists of building up clear and most general possible definitions for the student. Discussions of specific, concrete, and intuitive mathematical objects is avoided, or reserved for homework exercises, Instead, a focus on very general axioms that can apply to many specific structures of interest is favored.

The Russian method, in contrast, stresses specific examples and applications. The instructor gives specific, concrete, and intuitive mathematical objects and structures — say the integers — as a pedagogical examples of the abstract concept at hand — maybe rings, in this case. The student is given other specific instances of these general abstract objects as assignments — maybe some matrices, if we are looking at rings — and through exposure to many specific examples is expected to extract the formal axiomatic structure with which Bourbaki would have started. For the Russian, this overarching formalism becomes largely an afterthought; an exercise left to the reader.

As with many comparisons in education, neither method is strictly “better”. Nor should the names be taken as representative of the people that advocate for or are exposed to each method. For example, I am Russian but I feel like I learnt the majority of my maths following the Bourbaki method and was very satisfied with it. In fact, I am not sure where the ‘Russian’ in the name comes from, although I suspect it is due to V.I. Arnol’d‘s — a famous Russian mathematician from the second half of the 20th century — polemical attack on Bourbaki. Although I do not endorse Arnol’d attack, I do share his fondness for Poincaré and importance of intuition in mathematics. As you can guess from the title, in this article I will be stressing the Russian method as important to the philosophy of science and metamodeling.

I won’t be talking about science education, but about science itself. As I’ve stressed before, I think it a fool’s errand to provide a definition or categorization of the scientific method; it is particularly self-defeating here. But for the following, I will take the perspective that the scientific community, especially the theoretical branches that I work in, is engaged in the act of educating itself about the structure of reality. Reading a paper is like a lesson, I get to learn from what others have discovered. Doing research is like a worksheet: I try my hand at some concrete problems and learn something. Writing a paper is formalizing what I learned into a lesson for others. And, of course, as we try to teach, we end up learning more, so the act of writing often transforms what we learned in our ‘worksheet’.

Antoni Gaudi and learning algorithms from Nature

Happy holidays.

A few days ago, I was exploring Barcelona. This means that I saw a lot of architecture by Antoni Gaudi. His works have a very distinct style; their fluid lines, bright colours, myriad materials, and interface of design and function make for very naturesque buildings. They are unique and stand in sharp contrast to the other — often Gothic revival and Catalan Modernisme — architecture around them. The contrast is conscious; when starting out, Gaudi learned the patterns of the neo-Gothic architecture then in vogue and later commented on it:

Gothic art is imperfect, only half resolved; it is a style created by the compasses, a formulaic industrial repetition. Its stability depends on constant propping up by the buttresses: it is a defective body held up on crutches. … The proof that Gothic works are of deficient plasticity is that they produce their greatest emotional effect when they are mutilated, covered in ivy and lit by the moon.

His buildings, however, do not need to be overgrown by ivy, for Gaudi already incorporates nature in their design. I felt this connection most viscerally when touring the attic of Casa Mila. The building was commissioned as an apartment for local bourgeois to live comfortably on the ground floor off the rents they collected from the upper floors. And although some of the building is still inhabited by businesses and private residence, large parts of it have been converted into a museum. The most famous part among tourists is probably the uneven organic roof with its intricate smoke stacks, ventilation shafts, and archways for framing other prominent parts of Barcelona.

This uneven roof is supported by an attic that houses an exhibit on Gaudi’s method. Here, I could see Gaudi’s inspiration. On display was a snake’s skeleton and around me were the uneven arches of the attic — the similarity was palpable (see below). The questions for me were: was Gaudi inspired by nature or did he learn from it? Is there even much of a difference between ‘inspired’ and ‘learned’? And can this inform thought on the correspondence between nature and algorithms more generally?

I spend a lot of time writing about how we can use algorithmic thinking to understand aspects of biology. It is much less common for me to write about how we can use biology or nature to understand and inspire algorithms. In fact, I feel surprisingly strong skepticism towards the whole field of natural algorithms, even when I do write about it. I suspect that this stems from my belief that we cannot learn algorithms from nature. A belief that was shaken, but not overturned, when I saw the snake’s skeleton in Gaudi’s attic. In this post, I will try to substantiate the statement that we cannot learn algorithms from nature. My hope is that someone, or maybe just the act of writing, will convince me otherwise. I’ll sketch my own position on algorithms & nature, and strip the opposing we-learn-algorithms-from-nature position of some of its authority by pulling on a historic thread that traces this belief from Plato through Galileo to now. I’ll close with a discussion of some practical consequences of this metaphysical disagreement and try to make sense of Gaudi’s work from my perspective.

Pairing tools and problems: a lesson from the methods of mathematics and the Entscheidungsproblem

Three weeks ago it was my lot to present at the weekly integrated mathematical oncology department meeting. Given the informal setting, I decided to grab one gimmick and run with it. I titled my talk: ‘2’. It was an overview of two recent projects that I’ve been working on: double public goods for acid mediated tumour invasion, and edge
effects in game theoretic dynamics of solid tumours. For the former, I considered two approximations: the limit as the number n of interaction partners is large and the limit as n = 1 — so there are two interacting parties. But the numerology didn’t stop there, my real goal was to highlight a duality between tools or techniques and the problems we apply them to or domains we use them in. As is popular at the IMO, the talk was live-tweeted with many unflattering photos and this great paraphrase (or was it a quote?) by David Basanta from my presentation’s opening:

Artem: I don't follow the "pick the best tool for the job" mentality 1/2 @mathonco

— David Basanta (@dbasanta) February 23, 2015

Artem: I rather pick the best goal with the best tool 2/2 @mathonco

— David Basanta (@dbasanta) February 23, 2015

Since I was rather sleep deprived from preparing my slides, I am not sure what I said exactly but I meant to say something like the following:

I don’t subscribe to the perspective that we should pick the best tool for the job. Instead, I try to pick the best tuple of job and tool given my personal tastes, competences, and intuitions. In doing so, I aim to push the tool slightly beyond its prior borders — usually with an incremental technical improvement — while also exploring a variant perspective — but hopefully still grounded in the local language — on some domain of interest. The job and tool march hand in hand.

In this post, I want to unpack this principle and follow it a little deeper into the philosophy of science. In the process, I will touch on the differences between endogenous and exogenous questions. I will draw some examples from my own work, by will rely primarily on methodological inspiration from pure math and the early days of theoretical computer science.

Five motivations for theoretical computer science

There are some situations, perhaps lucky ones, where it is felt that an activity needs no external motivation or justification.  For the rest, it can be helpful to think of what the task at hand can be useful for. This of course doesn’t answer the larger question of what is worth doing, since it just distributes the burden somewhere else, but establishing these connections seems like a natural part of an answer to the larger question.

Along those lines, the following are five intellectual areas for whose study theoretical computer science concepts and their development can be useful – therefore, a curiosity about these areas can provide some motivation for learning about those cstheory concepts or developing them. They are arranged from the likely more obvious to most people to the less so: technology, mathematics, science, society, and philosophy. This post could also serve as an homage to delayed gratification (perhaps with some procrastination mixed in), having been finally written up more than three years after first discussing it with Artem.

What makes a discipline ‘mathematical’?

While walking to work on Friday, I was catching up on one of my favorite podcasts: The History of Philosophy without any Gaps. To celebrate the podcast’s 200th episode, Peter Adamson was interviewing Jill Kraye and John Marenbon on medieval philosophy. The podcasts was largely concerned with where we should define the temporal boundaries of medieval philosophy, especially on the side that bleeds into the Renaissance. A non-trivial, although rather esoteric question — even compared to some of the obscure things I go into on this blog, and almost definitely offtopic for TheEGG — but it is not what motivated me to open today’s post with this anecdote. Instead, I was caught by Jill Kraye’s passing remark:

[T]he Merton school, which was a very technical mathematical school of natural philosophy in 14th century England; they applied mechanical ideas to medicine

I’ve never heard of the Merton school before — which a quick search revealed to be also known as the Oxford Calculators; named after Richard Swinehead‘s Book of Calculations — but it seems that they introduced much more sophisticated mathematical reasoning into the secundum imaginationem — philosophical thought experiments or intuition pumps — that were in vogue among their contemporaries. They even beat Galileo to fundamental insights that we usually attribute to him, like the mean speed theorem. Unfortunately, I wasn’t able to find sources on the connection to medicine, although Peter Adamson and Jill Kraye have pointed me to a couple of books.

Do you have pointers, dear reader?

But this serendipitous encounter, did prompt an interesting lunchtime discussion with Arturo Araujo, Jill Gallaher, and David Basanta. I asked them what they thought the earliest work in mathematical medicine was, but as my interlocutors offered suggestion, I kept moving the goalposts and the conversation quickly metamorphosed from history to philosophy. The question became: What makes a discipline ‘mathematical’?

A year in books: philosophy, psychology, and political economy

If you follow the Julian calendar — which I do when I need a two week extension on overdue work — then today is the first day of 2015.

Happy Old New Year!

This also means that this is my last day to be timely with a yet another year-in-review post; although I guess I could also celebrate the Lunar New Year on February 19th. Last year, I made a resolution to read one not-directly-work-related book a month, and only satisfied it in an amortized analysis; I am repeating the resolution this year. Since I only needed two posts to catalog the practical and philosophical articles on TheEGG, I will try something new with this one: a list and mini-review of the books I read last year to meet my resolution. I hope that based on this, you can suggest some books for me to read in 2015; or maybe my comments will help you choose your next book to read. I know that articles and blogs I’ve stumbled across have helped guide my selection. If you want to support TheEGG directly and help me select the books that I will read this year then consider donating something from TheEGG wishlist.