Arstechnica (hey, they’ve had some back-to-back articles I want to comment on!) has posted an article discussing a panel talking about the Limits of Understanding. Particularly, the panel seemed to focus on apparent deficiencies in math – notably, the practical intractability of biology to mathematical modeling and Godel’s Incompleteness Theorem.

Now, the proper modeling of biology may simple be a problem of time…sometime, a mathematical model may be devised of sufficient complexity and rigor that it is considered an adequate analysis of the underlying systems leading to observable reality. What exercises the mathematicians more is whether an “adequate model” is a true model, that is.

The statement which interests me the most, though, is the final two sentences, uttered by the author of the article himself:

If math turns out to be just a tool (and a tool with some substantial limits), that may disappoint mathematicians, but it won’t necessarily slow down our ability to understand and model the natural world. This may be my background as a scientist talking, but that seems like the most important consideration, and I’m willing to live with a community of disappointed mathematicians in order to get there.

This demonstrates an infuriating philosophical stance. A model does two things: it relates observations to one another, expressing a causal relationship chain between them, and elucidates the underlying real factors which produce these observations. There exist two parts to the natural world: the observable, particular parts and the invisible relations underlying them. Two rocks are observable, the force holding them to the ground, or the force value we say they exhibit as a property is not. We infer force, based on changes over time. Force isn’t so much real as a very successful description of an inferred relationship. Mass is similar: it’s a relationship one objects exhibits vis a vis all other possible objects. Importantly, mass was very successfully describes as a constant property of a given set of material. However, mass can equally well be described as a property of curved space surrounding an object (if we’re talking about the source of the “attractive” force) or as a measure of the type of curvature produced by a massive object.

The statement by the author above exhibits a certain lack of philosophical follow through. A model which successfully relates observed experience to one another implicitly makes statements about the invisible relationships observations have. We cannot “understand and model” the natural world correctly unless we also correctly model those invisible relationships. Unfortunately, those relationships are invisible; the best we can do is infer their nature. If we have inferred incorrectly, then we are wrong, and our understanding is flawed. We DON’T understand. Since these relationships are invisible, this is somewhat untestable, so we won’t know. The hope is that, by using rigorously consistent systems, systems which exhibit the same properties as the fundamental relational properties of existence, we can leave behind some of those inferential worries.

What is irritating the mathematicians is that Godel seems to have annihilated that possibility. What really frustrates them is that if Godel truly killed the possibility of a formal system being provable, or that it is composed of two distinct subsets of theorems: those which are provable and those which are not, then do the relationships underpinning the universe dividable into a set of provable relations? That is, do all the relationships of the universe derive from a rational order. Importantly, Godel showed that certain theorems of a given system cannot follow from an entire set of other theorems. It’s like saying a relationship could be true, or false, and it bears no relation to other theorems. That’s fine, you might say, we can just look and see, right?

That’s part of the mathematicians problem, though: you can’t. We’re speaking here of the fundamental relations of the universe, not the observable particulars. Physics and biology aren’t much use if they can’t predict what’s going to happen, based on other things it knows. Godel showed that, if you have a formal system of rules that is good enough to describes math, then there are going to be statements in that system YOU CAN’T PROVE. Which means you can’t predict. If someone asks, you have to say “I don’t know”. Hofstadter shows (in *Godel, Escher, and Bach*) that there exists an infinite set of such statements.

If we take the positivist approach of simply using math as models where appropriate, we’ve thrown up our hands and admitted: we can’t actually describe the universe. We can describe bits of it, but we can’t relate it together. Further, because we cannot assert that the universe is describable by fundamental, consistent laws – because there cannot exist any such systems, including the universe – we can’t actually say the models we use are true, correct descriptions of how the universe works. That’s not decidable. All we can say is they work. There demonstrably exists at least one other formal system which can accurately describe the observable phenomena a model describes, and it is impossible to decide between them. In fact, it’s not even possible to say the universe is rational, because that, too, is undecidable.

If the purpose of the University is the search for truth, then it becomes problematic for the University if it can be rationally demonstrated that Truth is unreachable. That means its purpose is finished: it has found the one Truth: there is no Truth. Everything else is mental masturbation.

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## Quantity and Quality: Thoughts on the Relevance of the Number of Publications From Career Scientists

Posted in Academia, Commentary, tagged Academia, Church of Reason, Science, Universities on June 6, 2010| 1 Comment »

I came across an article on Arstechnica about studying the relationship between ‘rising stars’ in academia and mentors. I’m not terribly interested in talking about the specifics of the paper cited (particularly given I haven’t read it and instead am relying on Ars’ summary being sufficient for comperhension). Instead, I want to point out this quote:

Fecundity is simply a measure of the number of mentee grad students a professor advises throughout their career.

Academic success seems like a difficult to define term, because the University doesn’t have a goal which is easily quantified. That is, an academic is successful if they help the University increase its understanding of the rational world. The Church of Reason is desirous of the improvement of the rational understanding it has of the universe and making sure that understanding is tended and kept healthy. Obviously, this involves the increase of knowledge, but knowledge is really something of a secondary goal here. Knowledge, as I’m meaning it here, is the demonstration that a particular proposition holds true. Most knowledge is trivial; for instance, the record that a thermometer measured the temperature today in Winnipeg is of some scientific interest, but isn’t a high-quality bit of knowledge. High-quality knowledge comes in the form of general propositions which form the basic nodes in the big knowledge graph of Science. Where temperature readings are the high ceiling of the University’s edifice, general propositions are its columns and foundation.

However, it takes a certain amount of creativity and an extraordinary amount of thought to derive a general proposition. Consider the contributions of the giants of physics – men like Leibniz, Newton, Euclid, and Einstein. How many pillars did these great minds erect in the temple of science? It is an extraordinary feat to offer up an entire system of thought for the analysis of some major section of scientific knowledge…just one such offering puts one’s name on the canvas of history for centuries. How many major publications would this involve? How many papers per year?

How much time would be lost in the process of getting a paper to publication, time which could have been spent not being distracted from whatever the great work is? It feels odd to me that the number of publications one has to one’s name might be called a measure of academic success, when it seems highly likely that these papers, when delivered in such quantity, are highly likely to deal with the leaves, rather than the roots of knowledge. That’s important, sure…but I feel like it completely misses something important.

Obviously, the entire notion of fecundity, mentorship, and even election to the NAS seems to leave out the lone wolf. Science is now a team effort. If I can mentee a grad student, why, I am privy to their general direction of research. Each grad student I advise also advises me, and I can take quiet advantage of that, plus the opportunity to direct their labor in support of my own. My success in mentoring a student who then goes on to succeed establishes a relationship I’d nearly be remiss not to at least maintain, leaving me with a quiet advantage in my research efforts. It reminds me, when it comes right down to it, to human politics everywhere. That comes as a bit of a letdown, I guess.

For instance, in philosophy, prolific publication doesn’t seem to correlate well with quality of work (or lack). Wittgenstein published just two treatises. Edmund Gettier published but a single, three page article which remains one of the most significant papers in modern epistemology. Plato and Nietzsche both published enormous amount of verbiage.

I suppose, ultimately, I just expect an academic to spend more time thinking and less time authoring thoughts for others. Oh well, I suppose if it works, it works.

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