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The Peculiar Faith of the Mathematician or Scientist
Saturday, April 3, 2010, 2:13 PM

I’m delighted to see that Freddie deBoer and Led-the-commenter have delivered excellent pushback on a weak point of my previous post. Their point in brief is that the mere fact that our contingent minds have had remarkable success in explaining the universe so far is no guarantee that they will continue to experience such success. Moreover, it is possible that there are deep fundamental questions that we will never even be able to posit, much less solve.

On these two points, I concede entirely. Both of them are correct! They are not, however, as disastrous for my argument that epistemic despair is unwarranted as they may at first seem. As I said before:

Contingent minds merely undermine the necessity of our being able to comprehend the world … they leave open, however, the possibility of contingent minds that “just happen” to be of the sort that can make sense of the universe in which they happen to be located. Nevertheless, Freddie is right about one thing: once we eliminate necessity, we need reasons to think that our minds are of the right sort;

My post was an attempt to provide such a reason. Not a proof, mind you, but suggestive evidence. It may help to consider my approach to be Bayesian. The mere fact that the sun has risen every day that I’ve been alive is not proof that it will continue to do so, but with a fairly modest set of prior epistemological commitments, it’s reason enough to think that my belief that the sun will continue to rise is justified. Similarly, the prior success of science in describing the universe is not a proof that fundamental reality is accessible to us, but it is, to me at least, strong evidence that our minds are not wholly disconnected from reality as some of the formalists and constructivists would have us believe.

Freddie’s and Led’s challenge still warrants investigation, however, and today is a particularly fruitful day on which to consider it. Today is Great and Holy Saturday, when our thoughts are drawn to the small band of disciples who along with Mary gathered outside the tomb of Christ, waiting and hoping for the resurrection of the Lord, their presence motivated by nothing more than a promise. What Freddie and Led have nicely pointed out is that mathematics and science are based on a similar kind of promise.

I recall another Saturday, several years ago, when I was in college and trying to decide whether to take the plunge and become a math major. Late that night, I ran into an inebriated grad student who, as it happens, was writing his dissertation on non-foundational set theory. The two of us chatted, and I explained my dilemma. His first question was blunt, in the manner of mathematicians: “Are you smart enough?”

“I think so,” I replied, “it seems like most of the hopefuls get weeded out by the first class that requires them to do abstract proofs, and I have no trouble with that, so I should be fine, right?”

He smiled drunkenly and shook his head. ”No, proving things is the easy part.”

He was right of course, the difficulty of proof pales in comparison to the difficulty of stating what you wish to prove. Mathematicians since well before Hardy have been publishing paeans to proof as a creative and intuitive process, but trying to determine which questions are mathematically interesting is a far more daring act.  The aesthetic and analytic faculties must operate in full concert, fueled by the belief that what seems like it should be true actually is true… and provable.

Mathematicians have struggled with these doubts ever since Gödel showed that all that is true is not provable and, more importantly, since Matiyesevich and Chaitin showed that many interesting true statements are unprovable, rather than just Gödel’s artificial corner-cases. Setting problems and working as a mathematician, however, requires a further faith — a faith in the overall coherence of mathematics and in our ability to apprehend it.

I think the proper scientific analogue is nicely raised in Max Tegmark’s excellent paper on neo-Platonism. In order to work, the physicist must believe that we do not reside within the “physics doomed” quadrant of the diagram on page 12 of that paper. The point is that physics and mathematics are both epistemologically daring activities. I’ll  hasten to add that this in no way implies the truth or validity of the particularly bold prior commitments that the physicist and the mathematician hold, consciously or unconsciously. Freddie and Led have done us a service by reminding us of just how non-foundational these enterprises are. They rest on strong basic beliefs about the nature of the universe and the nature of our minds.

The inevitable response, and one that I expect to see in the comments, is that philosophers of physics and philosophers of mathematics have come up with systems within which these activities make sense even if they are divorced from Truth. Some of these systems even give explanations for the observed coherence, consistency, and success of these fields without making any appeal to correspondence with reality.

This is entirely true, and I won’t contest it, what I will say is that however successful these systems are philosophically, they are laughably out of line with the psychology of actual, practicing mathematicians and scientists. Anecdotally, I have never met a mathematician who, when asked what he does for a living, says: “I shuffle formal symbols in arbitrary patterns that are internally consistent and make sense to me.” Nor have I met a physicist who would reply: “I make tautological statements about internal questions related to the socially constructed version of reality that I’ve received.”

Is it psychologically likely that somebody who holds such beliefs would go through the trouble of doing mathematical or scientific research? I doubt it, in my experience they tend to become philosophers of science or mathematics instead. For a less anecdotal take on this, I’d recommend Feyerabend and Norris.


Can Reason Withstand the Death of God?
Friday, March 26, 2010, 6:33 PM

Freddie has written a post that forces me into the odd position of defending Sam Harris; the crux of which is the claim that once we accept the human mind as being a contingent accident of evolution, we necessarily must abandon any faith in the intellectual edifices constructed by such minds:

For me, I would merely put it this way: that we do not encounter the physical universe unmediated but through a consciousness mechanism and sensory inputs that seem to be the products of  evolution. And the belief (however you want to define a belief) in evolution makes the idea of those consciousness and sensory mechanism being capable, no matter how long the time scale, of perfectly or non-contingently ordering the universe around us seem quite low. Evolution does not produce perfectly fit systems, it only eliminates those systems so unfit that they prevent survival and the propagation of genetic material. A chimpanzee’s intellect is a near-miracle, capable of incredible things, but it will never understand calculus. I could never and would never say this with deductive certainty, but it seems likely to me that our consciousness has similar limitations.

Now, far be it from me to to diss Nietzschean perspectivism (I am, after all, on record as being an intractable opponent of the Invisible Eye), but I think Freddie overplays his hand here. Contingent minds merely undermine the necessity of our being able to comprehend the world (a necessity that the faithful take quite seriously, as an old Dominican friar once explained to me), they leave open, however, the possibility of contingent minds that “just happen” to be of the sort that can make sense of the universe in which they happen to be located. Nevertheless, Freddie is right about one thing: once we eliminate necessity, we need reasons to think that our minds are of the right sort; after all, the humble Giraffe is well adapted to its environment, but will never come to understand particle physics or the workings of its own neurophysiology. How are we to know that we are not like Giraffes, only with considerably wider possible-knowledge horizons?

A simple response is that we haven’t failed yet. The theories we build in order to explain the universe around us are remarkably, even distressingly successful. Even stranger than their success is the methodology with which we go about building them. As Christopher Norris has beautifully documented, the positivist fairy-tale of open-minded scientists accumulating measurable evidence, making conjectures based on that evidence, and then seeking to refute those conjectures does not well describe the actual way that scientists operate. In fact, the process is a good deal more deductive — the vast majority of working scientists begin by assuming scientific realism, then asking what underlying, noumenal features of the world might lead to the kind of evidence that we observe, then building a theory concerning what other kinds of evidence these noumena might produce, then seeking confirmatory and disconfirmatory evidence.

If the world were actually non-objective, or even objectively real but of a kind that was inaccessible to our contingent reason, what would be the odds of this extraordinarily arrogant and presumptive process working — not just once, but over and over again, throughout human history? If mathematics were formalist or something akin to a logical game, then why would it be the case that sets of “toy” axioms rapidly turn out to be trivial or contradictory; while the axioms that seem to best model the world churn out theory after theory of incredible richness, whilst just barely shying away from having sufficient power to prove their own consistency, thereby rendering themselves inconsistent? Finally, why on earth do our mathematical theories and our scientific theories work so eerily well together? Why does Wigner’s “unreasonable effectiveness” exist?

Let us return to the giraffes! There is no evolutionary pressure to having minds that can figure out U(1) x SU(2) x SU(3) symmetry, or why it is that the spin of an electron has to be what it is (also due to symmetry constraints). Freddie might reply that the ability to perform the kind of abstraction and symbolic thinking that is useful when figuring out how to hunt or how migration patterns work leads very naturally to the kind of abstraction required to figure out particle physics, but I think this is missing the point. The question is why fundamental physics is amenable to this kind of abstraction. Why minds of our kind happen to be in a universe of this kind. The alternative is not necessarily chaos.

I’ve occasionally been fond of saying that physics might be hopeless. Recall that a giraffe is well adapted to its environment, but will never figure out the fundamental properties of the universe. Similarly, physics could be trivial — it would be if we were supermen with superbrains.

So let us turn that argument around. Suppose that there are evolutionary constraints on the kind of intelligence that can arise, such that in broadest strokes, thinking, abstracting beings that evolve will tend to think like us and not like giraffes or superbrains. I’m still intrigued by why it is that fundamental physics is not amenable to giraffes (and hence trivial for us) and not challenging for superbrains (and hence impossible for us).

In fact, if one were to take my argument further, one could almost turn it into a new fine-tuning argument. It is  an observable, contingent, historical fact that our minds are of just the right kind to be able to figure out a great deal about the universe, while keeping the figuring-out process challenging. In the space of possible world with contingent minds, that seems very unlikely. Might we take this as evidence that our minds are, in fact, non-contingent? Might the success of science give us a reason for faith?

I’ll leave Freddie to answer that one. The argument I’ve outlined above is certainly hand-wavy, and needs fleshing out. In fact, I can immediately think of two replies:

1) Fundamental physics may, in fact, turn out to be impossible for us to figure out.

2) The Turing attack: Anything a superbrain can do, we can do too. It’ll just take longer.

I believe I have responses to these replies, but I’ll hold off on employing them until necessary.

Don’t let me give you the impression that epistemic despair is the whole substance of Freddie’s post. There’s a good deal more in there, including a blistering attack on “totalizing” moral realism that seems designed to stir this blog into action. I’ll leave a reply to that thread, however, to James and the others… for now.


Onwards! Ever Onwards!
Wednesday, March 24, 2010, 1:46 AM

Via Julian, I see that Yglesias has spun a narrative:

For the past 65-70 years—and especially for the past 30 years since the end of the civil rights argument—American politics has been dominated by controversy over the size and scope of the welfare state. Today, that argument is largely over with liberals having largely won.

…

The crux of the matter is that progressive efforts to expand the size of the welfare state are basically done.

This seems plausible, but I think I can spin an equally plausible narrative of my own:

Imagine, in a parallel universe if you prefer, that some of Yglesias’ ideological compatriots are driven not by wonkish numeracy, but rather by a deep-seated desire to abolish suffering. Like the War on Terror, the War on Suffering (declared June 1st, 2017) can never be won; but for this hardened, elite team of poverty-warriors, it must eternally be fought. More and more resources are tossed at the task, the cost/benefit ratio spirals higher and higher, but the grim band presses on, for press on it must, for to do any less would be to admit defeat.

One dark and stormy night, a happiness researcher knocks on the door of the Department of Pleasure: “Did you know that our metrics suggest that happiness is sometimes a function not of absolute wealth but of relative wealth? On the other hand…” ENOUGH! The staff at the Department of Pleasure (formed by the Providing America Tools for Hastening the Obliteration of Suffering Act of 2037) are uninterested in the effects of culture upon the human psyche or the dubious methodology underlying the findings. Somebody out there feels bad about the fact that his neighbor has more resources! So quick! To the Bat Cave! After many quests and trials, our triumphant heroes manage to rescue the Phoenix Sword from the Dragon King, and bring its enormous, equalizing, leveling forces to bear upon society. Finally, and at long last, we will have discovered happiness.

Pure speculation, I know, but is it so far from the mark? After all, as we know, “one is too many”.

Kidding aside, I’m not grokking the obvious footholds on this slippery slope. Is my read on the motivations flat-out wrong? It seems that much of the tradition of political philosophy from which modern advocates for the welfare state draw their sustenance has the abolition of suffering as its explicit goal. I wish I could be confident in Yglesias’ read of his movement, but I see no reason why, absent tactical considerations, they should be satisfied with Obamacare.


Whither the Post-Healthcare Realignment?
Wednesday, March 24, 2010, 1:02 AM

Over at Cato, Julian Sanchez has written a post about how the aftermath of healthcare reform could reveal faultlines in existing political coalitions and trigger realignment:

There’s no intrinsic commonality between, say, “left” positions on taxation, foreign policy, and reproductive rights—the label here doesn’t reflect an underlying ideological coherence so much as the contingent requirements of assembling a viable political coalition at a particular time and place.  If an issue that many members of one coalition considered especially morally urgent is, practically speaking, taken off the table, the shape of the coalitions going forward depends largely on the issues that rise to salience.

My excerpt doesn’t do justice, read the whole thing. I’ll revisit the last paragraph since it’s particularly important.

For now, however, let me point out that there are two broad ways of looking at the underlying motivations of political actors. The first view, one which I give little credence to, understands individuals as being motivated primarily by “issues”. That is, discrete views on a set of minimally overlapping topics to which we give greater or lesser importance in our internal rankings. This is certainly an accurate view when applied to certain subsets of the population. Professional lobbyists, for instance, are likely to have their activities driven by their views on a particular issue.

We generalize too easily beyond the subset for which this view is appropriate, however; and I suspect that this generalization is driven in part by the tools for measuring political attitudes that we have at our disposal. Polls tend to ask about issues, people respond to polls with statements about issues, and consequently we come to believe that issues predominate in general. This is not necessarily accurate.

A deeper investigation of political motives might reveal that issues are epiphenomena of worldviews — sometimes better and sometimes worse thought out, but always holistic, visions of what the ideal society would look like. Worldviews tend to be nebulous, they tend to be conveyed easily through imagery and poorly through scales that range from “Agree Strongly” to “Disagree Strongly”, and most of all they tend to resist discretization. I suspect that motivation based on worldview is overrepresented among professional political activists, think tank employees, and journalists.

So far so good, nothing I’ve said precludes realignment. It’s doubtless likely that many who support a particular contingent ideological ensemble share very different worldviews that happen only to align on key issues. What changes the picture is the fact that the political agenda is largely set by a self-selecting group of professional activists, think-tank workers, and journalists. Within this environment, pressure to be a “team-player” for your coalition is high. So too is the institutional inertia embodied by foundations, policy shops, and alliance umbrella-groups. Given this, one might expect the worldviews of the key players to undergo drift, to slowly mutate until they are more homogeneous and more compatible on all fronts. In fact, we see this occur quite frequently: one example is the labor movement, which for a long time was considered hostile to immigration, but which now is mostly in favor of comprehensive immigration reform. I suspect that this shift percolated down from the top, from the professional ranks of DC labor activists, who in turn underwent internal shifts in worldview due to the pressures exerted upon them by the contingent alliance in which they found themselves.

If this hypothesis is correct, we would expect political coalitions to be “stickier” than they would otherwise be. The fact that the Great Conservative Crack-Up that so many predicted after the fall of the USSR still hasn’t fully arrived might be considered evidence in favor of this hypothesis. One testable prediction might be that the rift between scientism and romanticism won’t immediately fracture the left after global warming ceases to be an issue.

What might melt the stickiness, however, is the systematic undermining of the intra-alliance power of the political professionals, as Julian describes:

…the possibility that I find interesting is that—against a background of technologies that have radically reduced the barriers to rapid, fluid, and distributed group formation and mobilization… They’ve already shown they’re capable of surprising alliances—think Jane Hamsher and Grover Norquist.  … It’s entirely possible that there are latent and dispersed constituencies for policy change outside the bipartisan mainstream who have now, crucially, been connected: Any overlap on orthogonal value dimensions within or between the new groups won’t necessarily be evident until the relevant values are triggered by a high-visibility policy debate.

It seems like you need something else to make this claim: the assumption that all these cool, new, distributed advocacy groups don’t get sucked into the extant ideological groupings over time. So long as we have the two-party system, incentives to conform are high and tipping points are rare but dramatic.

As for the Hamsher/Norquist folie à deux, such things aren’t exactly unprecedented, nor are they only a product of the digital age. Recall Karl Hess, or perhaps Gus DiZerega, who in 196Z was president of his university chapters of both SDS and YAF. What did he do after that? Why, he blogs about Paganism of course.


“The Unsustainable Autonomy of Reason”
Sunday, February 7, 2010, 8:25 PM

When was the last time you heard a transhumanist say something like this?

…the Enlightenment project of Reason to which many transhumanists are committed is self-erosive and requires nonrational validation. Transhumanist advocates for Bayesianism and transcending cognitive biases need to confront the repeated implosions of the religion of Reason into romanticism and mysticism, and develop more sophisticated and nuanced defenses of rationality.

If you’re as pleasantly shocked as I am, click over to the Institute for Ethics and Emerging Technologies, where Prof. James Hughes is churning out a scrupulously fair and even-handed series of posts entitled “Problems of Transhumanism”.


Beyond Möbius Bagels: Twisty Bagels
Tuesday, December 8, 2009, 11:19 AM

Rarely do Joe Carter and MAKE Magazine point me towards the same online curiosity. The intersection of topology and breakfast must have considerable ecumenical appeal. I’m going to turn Hart’s challenge around, however: now that you know how to create two interlocking bagel-halves by performing a twisty cut on a simple space, I’d like you to try to produce something topologically equivalent to two interlocking bagel-halves by performing a simple cut on a twisty space.

Here’s a hint:

Imagine that I have unrolled a Möbius strip. I’ll get something that looks like this:

Consider and unrolled Möbius strip.

It should be easy to see how we can recreate the original Möbius strip by gluing together the top and bottom edges (marked ‘A’) in such a way that the arrows align.

Now consider a line drawn around the circumference of the strip. On our unrolled strip, it will look like the dashed line ‘B’ below:

WillMöbiusStripAsSquare2

I will state but not prove that there is no way to continuously deform the line B to form a new line B’ which both also forms a complete loop wrapping around the strip AND does not intersect our original line B at any point. This should be intuitively quite obvious.

Now, what does this tell us about the space(s) that will result from cutting the Möbius strip along line B? Generalize what you’ve found to tori and you will have discovered the Twisty Bagel.

What have we learned? That sometimes the cut is relative to the space. Determining whether that is always the case would require clarification of our hazy concepts “cut” and “space”.


Which is more annoying?
Monday, December 7, 2009, 2:28 PM

Which is more annoying: Leftists who in the wake of Climategate have suddenly discovered a love for public choice theory or libertarians who in the wake of Climategate have studiously ignored public choice theory?

Whatever your answer to that question, I hope it suggests that public choice theory is not a complete description of the world, and that ideology remains a powerful influence on the decisions of politicians.

P.S. This post should in no way imply that all leftists and all libertarians have behaved disingenuously.


Unity through Beauty
Friday, December 4, 2009, 12:23 AM

Longtime readers know of my obsession with mathematical beauty, so it should come as no surprise to find me hopping up and down most eagerly and pointing you towards Matthew Milliner’s very immodest proposal in Public Discourse. My only quibble with the article is that the proportion of mathematicians I know who view their field as “an escape from religious questions” is vanishingly small.

While you’re doing that, I’ll be putting Matthew’s suggestions into practice. You see, a long-awaited copy of Tafeln Höherer Funktionen arrived today, courtesy of Mr. Daniel Viertel of Leipzig, so I’ll be spending the next couple of days enmeshed in it. Here’s a little sample, taken from page 13:

GammaThere’s something so unspeakably charming about hand-drawn graphs. I should hope that some enterprising individual has written a chunk of code that can turn sterile computer-generated curves into faux-authentic computer-generated curves, but if it hasn’t happened yet then you all should consider this to be a formal request.


The Neverending Loyalty Debate
Wednesday, December 2, 2009, 12:10 PM

Responding to Helen, Conor fails to acknowledge the distinction between “critiquing an argument” and “writing a hit piece”, or at the very least implies that the two phrases may be used interchangeably.

It’s funny, I thought Helen’s post made it pretty clear that she was not objecting to changing one’s mind, or to writing about how one has changed one’s mind, but rather to writing about it in a certain scummy, sniveling, skulking, yet sanctimonious way. In other words, I read Helen’s post as making a partially aesthetic claim rather than a purely positive one, with the positive subclaim restricted in scope. Then again the two are frequently combined in Helen’s writing, so who knows?

In any case, I think Conor is being rather unfair in broadening what was a narrowly-argued point, but my undying loyalty to Helen may be clouding my judgment.

P.S. Previous installments in this saga may be found here, here, here, and finally here.


Sayonara, Weber!
Wednesday, December 2, 2009, 8:18 AM

Via Tyler Cowen, a paper by Davide Cantoni  casts some doubt on the efficacy of the Protestant Ethic:

Many theories, most famously Max Weber’s essay on the ‘Protestant ethic,’ have hypothesized that Protestantism should have favored economic development. With their considerable religious heterogeneity and stability of denominational affiliations until the 19th century, the German Lands of the Holy Roman Empire present an ideal testing ground for this hypothesis. Using population figures in a dataset comprising 276 cities in the years 1300-1900, I find no effects of Protestantism on economic growth. The finding is robust to the inclusion of a variety of controls, and does not appear to depend on data selection or small sample size. In addition, Protestantism has no effect when interacted with other likely determinants of economic development. I also analyze the endogeneity of religious choice; instrumental variables estimates of the effects of Protestantism are similar to the OLS results.

Does anybody else appreciate the irony of the above quoted paragraph? Remember, this is Weber that we’re talking about.

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