I just started reading an interesting book, "How Mathematicians Think," written (naturally enough) by a mathematician (William Byers). It got me thinking not only about mathematics but also science, what it is and why I do it. Here's the paragraph that triggered it:
The most pervasive myth about mathematics is that the logical structure of mathematics is definitive--that logic captures the essence of the subject. This is the fallback position of many mathematicians when they are asked to justify what it is that they do: "I just prove theorems." That is, when pressed, many mathematicians retreat back to a formalist position. However, most practicing mathematicians are not formalists: "what they really want is usually not some collection of 'answers'--what they want is understanding [cite is to an interview with mathematician William Thurston]." The statistican David Blackwell is quoted as saying, "Basically, I'm not interested in doing research and I never have been. I'm interested in understanding, which is quite a different thing." (Byers, How Mathematicians Think, pp, 25-26; emphases in original)
I read this and my first reaction was disagreement, although I couldn't figure out why. The obvious reason was that I consider myself a researcher and I am interested in research. But I'm also interested in understanding, and I didn't see those things as in opposition, although I had to agree that doing research and understanding are different, so there was a problem somewhere. When people ask what moves me to do science, I almost always say something like, "I want to find out how the World works."
I like to think of myself as an altruistic person and naturally I hope that what I do will make the world a better place for everyone. That does direct my curiosity to some extent. I'm not indiscriminate about what parts of the world's workings I'm interested in. But if I am completely honest with myself (and like everyone I'm probably often not honest with myself) I think I'd say that the strongest motivator is just wanting to know the basic principles of the world I live in. That's more than just a description of the mechanism but the general plan which governs how the mechanism works. Like most scientists, I assume these principles are knowable, regular and that science is the way to know it.
None of that sounds incompatible with Blackwell's goal of "understanding," but there is one additional element that is crucial and goes beyond it. "Understanding" is something an individual has. It's subjective. It's a slippery concept, but it's like assimilating an explanation that satisfies us in some way. But if something is going to be scientific, the product of research, then it must be intersubjective. That's why religious or supernatural concepts aren't science for me. They can't be displayed in a way for all to see but depend on individual and subjective knowledge. The question isn't so much testability or falsifiability as it is what those things imply: the ability to make the evidence available to anyone and everyone. So when I say that research is trying to figure out how the world works, I mean producing explanations that are shareable and intersubjective, not just a form of my own understanding.
Science is a social enterprise, not a solitary one, even if some of us carry it on in a solitary fashion, alone with our computers, or paper and pencil. I do what some people might refer to as Grand Theory, but it is not meant to be Grand Theory just for me, but Grand Theory for the Ages. Pretty ambitious. But how else should we do it?
We all know that hard won science is in a sense temporary. That at some point a revision will come along that will sweep away what was considered true in the past. It doesn't matter. I'm still trying to build something that will last forever because it's True. We all lie to ourselves. Even when we think we are being honest with ourselves. Go figure.
Interesting post, holmes.
W. Boyd Stelzner, EE chair at U of Arkansas during the transitor/early atomic energy eras -- my grandfather, said: Scientific discoveries are not new knowledge. All knowledge exists now, but we do not understand much of it yet. Our role as a scientist is to organize what is currently known so our students will understand more.
We all know that hard won science is in a sense temporary. That at some point a revision will come along that will sweep away what was considered true in the past.
I think it's a good enough achievement to be able to share with others, if only for a while, the best current approximation to reality.
However obscure one's field, if another person builds upon one's work and thereby furthers human understanding, one can smugly (but quietly) claim to have the shoulders of a giant.
I've been following and appreciating your blog for several years, and I'm very sorry to see (a few posts later) that you're closing up shop. I've never before commented, but since you quote me within the quote above, I wanted to say a little while I still have a chance.
I haven't read Byers' book, but my essay which was quoted was intended to disrupt conventional assumptions, so I'm glad to see you fighting back --- that's an important part of the process.
Mathematics differs in significant ways from biology: it has internal standards for verification, which enables formal progress to be more rapid and clearcut (relative to personpower). People are is strongly tempted to think of mathematics as abstract, objective and detached. This is not the place to discuss mathematics, but despite the obvious differences, I think there is some parallel maladaptive behavior.
The key point is to accept what it means for us to understand. Human understanding is intrinsically human. It is very much tied to human experience and the architecture of our brains. We cannot, and should not even try (in the name of objectivity), to remove ourselves, our intuitions, and our feelings from our thinking and our conclusions. You in fact do not---your posts discussing epidemiology and comparing various kinds of clinical studies have been very nuanced and full of intuition and feeling. We need to be constantly vigilant and critical of ourselves as well as others, and constantly vigilant for mistakes coming from our assumptions and biases: there is no single objective external standard that can do it for us.
You say "Understanding" is something an individual has. It's subjective. It's a slippery concept, but it's like assimilating an explanation that satisfies us in some way.
Sure, an individual can have understanding. An individual can also have a virus, but that doesn't imply that a virus is a slippery subjective concept. Understanding can be communicated, and our job is really to reach better understanding and to communicate it. That's what you've been doing in your blog. Understanding is the way things lodge in our brains. It's because the biology of human brains is much alike that we can (in good circumstances) effectively communicate understanding provided we curtail our antiseptic measures. As you say, science is social. When people misunderstand or don't understand, they do not learn science.
You have a strong aversion to pseudo-science, religion, supernatural explanations. It's a mistake to let that aversion deflect you. You're in no danger of falling into those traps, and it's unnecessary and futile to demonstrate why you're more objective than they: to interested people of good will, it's clear.
Bill: First, I am stunned you read the blog. My son's PhD is in topology (thesis was on Coxeter groups) so your name and stature are very familiar to me. I don't disagree with anything you say, although I think I would have said it differently. But the main point I wished to make relates to what comes toward the end of your comment. I think you accurately diagnosed the connection to what bothers me about religious and supernatural explanations. But the emphasis here is on "bothers."
One of the big unsolved problems in the philosophy of science is the Demarcation Problem, i.e., what is the difference between science, pseudoscience and non-science. I think the non-science part is fairly easy. My mother was a professional artist (a painter) so I don't think everything in the world is science nor that science is the only way to "know" something (although what constitutes knowledge is again a thread I don't want to pull here lest everything become unraveled). The science/pseudoscience part, however, is much tougher. I don't consider astrology pseudoscience, I consider it bad science. It's just wrong. But saying how it's wrong (or if you are of another persuasion, pseudoscience) is much more difficult, and most philosophers of science I know have given up on the demarcation problem as not fruitful.
But I'm not a philosopher of science. I'm a scientist, so the question still bothers me. What I was trying to get at in that post was that one characteristic I am inclined to required from something that is scientific is intersubjectivity. "Understanding" is not intersubjective in my way of speaking but something confined to the understander. When understanding becomes intersubjective it makes a necessary step toward science. It seems not to be sufficient because religious understanding could also said to be intersubjective. That's when the shouting between foundationists, coherentists, verificationists, falsificationists and all the rest start. So while I am satisfied with intersubjectivity it doesn't get me all the way there.
I find Byers's book interesting but, so far as I've read, overstated. I like it because it makes me think which helps me devise strategies in my own research (which makes use of lattices) and It captures some of what I've seen in mathematics (I am far from being a mathematician but I was an undergraduate math major and had RH Bruck as my advisor; I learned analysis from Walter Rudin and logic from Stephen Kleene and I knew George Mackey well). But as much as I am bothered and sometimes baffled by what I do in my own science, I am even more so by "the unreasonable effectiveness of mathematics." It's a mystery to me, but one I am glad for.
At any rate, thank you for the kind words. Not blogging will feel strange to me, but it will give me a chance to do more of what I want to do, which involves mathematics. I have become interested in how the logic of inquiry in epidemiology has some analogies with observation in quantum mechanics and it isn't hard to show that the lattices generated by a typical data set in epidemiology (just a table of objects and their attributes but with the attributes having a certain structure like being partitions or linear orders) are non-distributive. The lattices come from what Birkhoff called a Galois connection and the induced closure systems and he and von Neumann used this in their paper on quantum logic in 1935. Unfortunately their lattices still have more structure than the epidemiological ones (they are orthomodular) and so far I'm not sure the same is true of the epidemiological kind. But I have a lattice theorist helping me so maybe I'll get somewhere with this. Or maybe not. Life is short and the world is mysterious and messy.
Anyway, thanks for the comment. To know that you read the blog blows me away. And makes me embarrassed about some of the facile explanations I've proffered.
It's been obvious to me that you have a strong affinity for and acquaintance with mathematics.
Let's not worry about the word "understanding" --- whether the word has a connotation of something private or something sharable is secondary. We both agree that we want "intersubjectivity". In my high school yearbook, I put as my goal "to understand", and I've thought of that as summing up what drives me.
My attitude toward the demarcation problem originated I think from childhood games my siblings and I used to play, where one of us would say something obviously implausible about the world, as if psychotic, and the others would try to trip up the fantasy and establish that it couldn't be right. We discovered how difficult it is to establish reality, and I started to think of these battles as futile. People of good will whose thinking is not confused and muddled or trapped in a rut can reach a common understanding. In the absence of good will or clarity, they do not, and an external criterion or external referee does not help. I think the only antidote to astrology, creationism and the like is a better communication of the understanding of how astronomy and biology actually work. Debates about astrology and creationism and the like are not effective, but when people better understand reality, they dissolve.
There is a counterpoint to "the unreasonable effectiveness of mathematics": the unreasonable difficulty people often have in catching on to mathematics. These are twin phenomena. Mathematics is about developing effective means of thinking. Mathematics is beautiful when we fit puzzling pieces together in our minds, into a coherent and simpler whole. Mathematics is effective because we have only one mind, no matter what we're thinking about. It's not so much the mathematical topic that is important---as Godel drove home, any part of mathematics really contains all of mathematics---it's how mathematics teaches us to think that is important.
When we acquire a pattern of mental organization and understanding that is effective for one subject, it is likely to be effective for others because our minds stay the same.
But the effectiveness depends crucially on how mathematics fits in our heads, something that people don't usually talk about or even consciously acknowledge. If I say "(x,y,z) satisfying x+y+z=1", someone may think of an equation, or an algorithm for plugging in numbers and getting solutions, or a plane in 3 dimensions that cuts a certain way through axes in a coordinate system (or a blank wall). The same formal mathematics can fit in our heads in many different ways, and when people discuss mathematics, these differences are usually (but not inevitably) invisible. I think it's important to bring them out, even though the psychology of mathematics may seem secondary and irrelevant and detrimental to air of objectivity.
Your idea about lattices and epidemiology is too cryptic for me to get a clear image --- I'm unfamiliar with the quantum logic stuff --- but I can imagine how it might be a helpful structure. The big problem of organizing and analyzing epidemiological information seems very interesting and challengingas well as important, and I wish you good success.
BTW, I was very taken by Kleene's book on Foundations of Mathematics when I was in college, and it motivated me to write a senior thesis on intuitionist topology. I thought I migh become an intuitionist logician, but when I approached Tarski to advise me, he said that Berkeley wasn't a good place for intuitionism, so I went into topology instead. I also used to think I would switch to biology when I reached the age of 35 or 40, because I was very drawn to the challenge of trying to understand life. It didn't happen.
Bill: You raise interesting and key points. If I think about your statement, " People of good will whose thinking is not confused and muddled or trapped in a rut can reach a common understanding. In the absence of good will or clarity, they do not, and an external criterion or external referee does not help," I wonder how we know when the premises are satisfied. After all there are large religious communities that could be said to consist of people of good will who have reached a common understanding. I would say their thinking is muddled but they wouldn't agree, so we are back to the criterion of muddled thinking or whatever we wish to call it. Astrology ha all the hallmarks of science. It has a theory, it makes use of empirical data, it is testable. It's just wrong. But then one could say the same of Newtonian physics.
As for mathematics, one of the theses of Byers's book is that the heart of mathematical thinking is not at all precision and disambiguation but ambiguity and paradox. He provides many examples of mathematical concepts (one is "a variable") that holds two contradictory aspects, one being any number in the domain, the other the specific number that gets plugged in. Your examples likewise are ambiguous, which you express by saying they can fit in our minds in different ways. Often we slide back in forth between those different ways as with Euclidean versus Cartesian plane geometry. I know I didn't explain my lattice interest in any way it could be understood if you didn't already know it (which is not much of an explanation for anything), but I mentioned it because the question of observation, measuring, ambiguity and trying to figure out the world's ontology (its furniture) is something I think about and how that particular post came about. The simple question about what it means for a thing to "have" a property arose as I was trying to rethink epidemiology from its foundations and it led me into very deep waters. Once that happened I started to become very nervous about some commonly expressed ideas about what science is and isn't. As one wag put it, "Expecting a scientist to understand what science is all about is like expecting a fish to understand hydrodynamics." We know how to do it, but we don't always know what we are doing or how what we are doing is different from something else.
At any rate, you might take a peek in Byers's book. You'll get the flavor after the first chapter, I think. It's a different perspective than yours. In the spirit of ambiguity (which he espouses) I see the value in each. There seems to be a paradox there somewhere. I'll have to Russell up an example.
BTW, my son says he spent many, many, many hours in the company of your mimeographed notes. I envy him the company he can keep. He's out of my league in that department, but I was thrilled that you read the blog. Now the blog is history. I'm still here but at some point I'll be history, too. That doesn't bother me.
The Demarcation Problem bothers me. I'm not sure what that indicates about my priorities.
I suppose I made too bold a claim when I said that people of good will whose thinking is not confused and muddled or trapped in a rut can reach a common understanding. There's an emotional dimension that is too culturally dependent, too powerful and too much a part of who we are for this to work in general. A counterexample is the Israeli-Palestinian conflict, so rooted in divergent cultural and emotional history that only time and history is likely to heal it.
But, when we're talking about something like evolution: it's a mistake to lump together all the different kinds of religious communities. Some of them consist of very reasonable people --- these are not the problem. You hedged your statement into the customary polite form, "religious communities that *could be said* to consist of people of good will". My take on the creationist camp is that the leaders are often charlatans, with varying combinations of self-delusion and cynicism, while the followers are often stuck in a rut through social and cultural pressure. They are in defense mode rather than truth-seeking mode, and I don't think it's worthwhile to engage them directly. But most people are probably in the middle, perhaps mouthing an opinion but not strongly locked to it. They haven't thought very hard or very clearly about the issue, and they are the ones who can potentially be educated.
Scientists also get into cultural / emotional ruts. One thing that comes to mind is the issues that surround the mental and emotional life of non-human animals, the debate about animal rights and the ways animals are used in research.
Concerning epidemiological deduction: are you familiar with "The China Study" by Campbell and Campbell? I'm curious about your take on it. (I felt somewhat frustrated that I didn't have the time or expertise to check out the primary data this book was discussing.) It has seemed obvious to me for some time that the traditional American truisms about nutrition were highly distorted by cozy relationships of the regulators with the agriculture industry. On the other hand some things I have read (e.g. some of John Robbins, Howard Lyman or Dean Ornish) seem too partisan in the opposite direction.
If this is getting to be too much of a private conversation, feel free to email privately.