Friday, October 05, 2007

Our Mathematical Universe

It was recently brought to my attention that Russell Howell, Professor of Mathematics at the Christian academy of Westmont College in Santa Barbara (California) has taken notice of my article "Fundamental Flaws in Mark Steiner's Challenge to Naturalism in The Applicability of Mathematics as a Philosophical Problem" (2003). He only quotes (in fact somewhat misquotes) a single isolated comment from my article and ignores almost everything else it says, even though the ignored material contradicts his argument in ways he curiously neglects to mention. His paper, "Does Mathematical Beauty Pose Problems for Naturalism?" was published in the 2006 issue of the online Journal of the Association of Christians in the Mathematical Sciences (yes, Christians have an association for everything).

I won't bother discussing the rest of his essay, since my article against Steiner already rebuts the same thesis Howell defends, and more than adequately in my opinion. I am only interested here in Howell's lame mischaracterization of my arguments, insofar as he quotes me at all.
Since I'm not the actual target of Howell's article, I'm only mentioned on page 9. He brings me up only when discussing Maxwell's use of a particular heuristic to discover electromagnetic radiation: by combining the assumption of a "conservation of charge" with mathematical descriptions of electrical systems that were already empirically established, he calculated (in effect) that energy should be leaking away from electrical systems (he was right: it was being converted into radio waves).

Howell is impressed that Maxwell did this by tinkering with mathematical descriptions, though as I argue in my article on Steiner, mathematics is nothing more than a precise language. It is thus no more amazing that tinkering with mathematical descriptions can discover the truth than tinkering with ordinary English descriptions can do so. Maxwell combined descriptions of conservation with descriptions of current and developed a new hypothesis worth testing. This is no different than combining "my wallet just went missing" with "pickpockets are often about" and developing therefrom a new hypothesis about what happened to my wallet. No one would conclude from this that the universe was anthropocentrically designed so I could discover pickpockets.

I suppose Howell misses this point because he thinks the word "mathematical formalism" conveys something special. But a "mathematical formalism" is still nothing more than a sentence in a language that describes an observed or hypothesized fact. It is only "formal" in the sense that it is descriptively more precise than, say, ordinary English. But it is otherwise no different. And insofar as anything that behaves consistently can be described using language, the fact that the universe can be described using the language of mathematics is simply a trivial consequence of the obvious, not a sign of anything deeply anthropocentric about the universe (as Howell might notice if he ever tried explaining how a universe would be incapable of being described mathematically). I say a lot more about these sorts of facts in my Critical Review of Victor Reppert's Defense of the Argument from Reason (2004) and in
my book Sense and Goodness without God (chapters II.2 and III.5.4).

But now back to the point...

 

Getting It Right

Here is how Howell (mis-)quotes me:

Maxwell rightly picked the simplest imaginable solution first, which due to human limitations is always the best place to start an investigation, and which statistically is the most likely [as] simple patterns and behaviors happen far more often then complex ones. [Thus] Maxwell's moves [that] anticipated EM radiation [were] therefore a natural conclusion from entirely Naturalist assumptions.
Howell immediately responds:
But with such language Carrier plays into Steiner's hands. Picking a simple solution in accordance with human limitations is precisely analogous to using the number ten as a means of unlocking secrets to the universe. It is quintessential anthropocentrism. Because of Carrier's background in history, one wonders if it is difficult for people who were not trained in science to appreciate how absolutely uncanny is the continued use of mathematical formalisms by physicists.
That's the sum of his argument against me. Let's begin with the fact that Howell altered his quote of me in ways he did not indicate. What I actually said was (putting the material he omitted now in bold):
Maxwell rightly picked the simplest imaginable solution first (e.g. that it all went one place, rather than several), which due to human limitations is always the best place to start an investigation, and which statistically is the most likely (simple patterns and behaviors happen far more often than complex ones—since Maxwell's day, again, the discoveries of Chaos Theory have changed that assumption, but again only after vast amounts of empirical evidence confirmed and thus justified the change in our assumptions).

That Maxwell's moves anticipated EM radiation was therefore a natural conclusion from entirely Naturalist assumptions. Charge was going somewhere, which we knew because the descriptions of charge behavior that we had, which were empirically well-grounded, left out and thus entailed the disappearance (or spontaneous appearance) of charge, which begged for an explanation. Maxwell hypothesized such an explanation by making some simple and obvious changes to the descriptions that accounted for this discrepancy--changes to the way the pattern of behavior was described that allowed inclusion of another element to that pattern. The changes he made were the simplest ones he could make that didn't invalidate but instead preserved the predictive success of the existing descriptions, while also bringing them into line with conservation laws. And the changes he made were still, in fact, hypothetical. They could have turned out wrong, and many tinkerings with these equations, by him and others, no doubt preceded this success and failed. But on Naturalism, his final guess was a smart one, and one likely to succeed. So we should not be amazed that it did.
With a closer look you can see how Howell's omissions (which he fails to indicate with ellipses or even address in his response) reveal how off-base his remarks are. To begin with, of course, we do not have to assume the universe is anthropocentric to believe in conservation of charge, or to suspect that the charge disappearing from our equations can most likely be recovered in our descriptions if one thing rather than several is responsible for the discrepancy. To start with the hypothesis that charge is disappearing from our equations due to a single phenomenon (like Maxwell's hypothesized "displacement current") is reasonable in any universe, whether anthropocentric or not, because a single explanation will always be more likely than several explanations just "happening" to cause the same effect at the same time. Such an assumption is also the easiest place for a human to start--hence that Maxwell would start there is fully understandable.

Howell conflates these two explanations into one, ignoring my statistical argument and pretending I only offered an argument from human limitations. He also gets wrong the import of both explanations, by ignoring what else I say, here and in the rest of my article. As one can see from the full and correct quote, I did not argue that Maxwell succeeded because he chose a "simple solution in accordance with human limitations" but that this human limitation is what caused him to try a simple explanation first. As I allude already in the quote above (and give specific examples of elsewhere in the same paper, which again Howell ignores), this tactic does not always work, a fact that actually refutes Howell's thesis. For instance, we "tried" a simple chemistry of four basic elements first. But it turns out there are over ninety.

The reason we tried four elements first is the same reason Maxwell tried to find only one cause first. The cause he was looking for was of the spontaneous appearance and disappearance of electrical charge in electrical systems as then described. Maxwell hypothesized that this 'disappearing' charge was never actually leaving the system, which entailed that, instead, energy had to be leaking from the system, in one way or another. And he hypothesized only one leak: which he called "displacement current," and which we now recognize as "electromagnetic radiation" (i.e. light and radio waves). But in chemistry this same tactic (of trying the simple solution first) failed to align with reality, as it often does not.

Hence it was entirely possible that charge was not conserved (after all, we now know matter is not), just as it was also possible that charge was being conserved but that energy was leaking from electrical systems in two completely different ways at the same time (or three or ten or twenty). Maxwell guessed it was one, and got lucky. But his luck is not surprising, since statistics favor the simple answer even in a blindly operating, undesigned cosmos--for obvious reasons: absent deliberate design, the more complex a system, the more improbable it is (as advocates of Intelligent Design are always reminding us). The improbable is not impossible, just less frequent, but that still means we will luck out more often if we start with the simpler hypothesis and work our way up from there. And though the causes of individual events are always incredibly complex, constantly repeating events are generally the result of the predominance of a few simple causes. Only a cosmic puppeteer could make it otherwise.

 

Reality Isn't Pretty

Thus Howell's contention that simple systems imply anthropocentrism is baseless--and in fact a little bizarre. Since his thesis entails we should expect an abundance of simple systems only in an anthropocentric universe, Howell apparently thinks if we found a completely unanthropocentric, undesigned universe, it would be fundamentally more complex than the one we are in. That certainly sounds perverse to me. And rather indefensible.

Nevertheless, there will always be complex systems, as simple systems will randomly and catalytically combine and interact even in an unplanned universe. In fact, most of reality is an immensely complex fabric of interacting systems, which individually are simple but in aggregate are not. However, since humans are really only good at solving the relatively simple problems, the reason we have discovered so many "simple" laws is that these are the kinds of laws we have most often been looking for, and are most able to find. Meanwhile, most of the universe is actually governed by "laws" so complex we have made little progress in predicting even commonplace phenomena, like earthquakes, or the weather, or even, in most cases, human behavior (a note to the nit freaks: I only use "law" here, and throughout, to mean a consistently repeating behavior of matter and energy, in keeping with modern metaphor).

Consider another passage from my article that Howell entirely ignored:

But isn't it at least the case that scientists have found a successful scientific method in focusing on 'beautiful' and 'convenient' mathematical theories? Not really. Though that has been an effective heuristic for getting at simple and focused problems in comprehensible ways, this is simply the result of human limitations: we have to start small, and solve simple problems first, in the few ways we know how and are best at. But if we were to rely solely on this heuristic, most of the greatest scientific discoveries would never have been made. Far from a "beautiful and convenient" chemistry of four elements, we discovered in the end an incredibly ugly, messy, and inconvenient Periodic Table of over ninety elements and counting (never mind the mind-boggling complexity of the Standard Model of particle physics); far from the "beautiful and convenient" planetary theory of Copernicus, the paths and velocities of the planets are so ugly and inconvenient that we need supercomputers to handle the messy intersection of Newtonian, Keplerian, Einsteinian, Thermodynamic, and Chaotic effects, and even then they are not always entirely accurate in their predictions on astronomical scales of time (like thousands and millions of years).
I think this argument should be extended beyond particular examples to the whole of science, and that is what I will do here, to make even clearer the argument I make against Steiner, and am making now against Howell.

Take Newton's formulas for motion and gravity (which some people inaccurately call "Newton's laws"). Many have thought these are beautifully simple (though in practice they typically require the application of calculus, a method of mathematical analysis so complicated many humans can't even learn it), but we should not let their "beauty" distract us from the fact that nothing in the real world obeys them. Even apart from the fact that Einstein found Newton's formulas needed to be much more elaborate and complex, and even apart from the fact that the laws of thermodynamics and quantum mechanics complicate the application of simple equations like Newton's to real-world cases, even setting all that aside, any competent scientist will tell you that if you run the same experiment several times, e.g. dropping an apple from a fixed height, you will get different results every single time. We only find Newton's laws of falling bodies in this discordant data by averaging experimental results out and rounding them off. Yet in reality, a falling apple will sometimes fall faster, sometimes slower, and this will be noticed more the more precisely you measure its fall.

Why? Because the world is an extremely messy, complex place. The moon's gravitational effect on a falling apple, for example, is constantly changing, as is the sun's gravitational effect, and Jupiter's, and so on, and even the earth's, as magma and continents and oceans and masses of air are always on the move, and even the rotation of the earth is always changing, while friction against the apple in the air will constantly change in response to variations in temperature and pressure, and even the apple's shape and mass will constantly change (as it gets dented from repeated dropping or squeezing, and emits olfactory molecules, and collects or sheds dust, and absorbs or evaporates moisture, and even as light bounces off of it, and cosmic rays pass through it, and now radio waves, and on and on), and so on (a complete list of variables would be immense).

Consequently, Newton's equations for motion and gravity only apply to ideal situations, which never in fact exist. That humans choose to focus on the ideal as a means to get a handle on the complexities of the real world is a product of human limitations. But this means Newton's laws are essentially human fabrications. We made them simple on purpose. Because we needed them simple to be useful. The universe, however, is never that simple. It never anthropocentrically conforms to our ideals. This does not mean there is no objective truth to Newton's laws. Rather, it means their truth is similar to that of Euclid's geometry. As we now know, there are non-Euclidean geometries, and in fact the real world obeys them far more frequently (another example of things turning out way more complicated than humans first thought). But Euclid's geometry often works well enough.

Why? Because Euclidean geometry is a description of what necessarily follows for any system that conforms to its axioms, as in fact Euclid logically demonstrated. So the more closely a real system fits those axioms, the more closely Euclid's "laws" will describe that system. His geometry thus becomes a useful tool, provided we are willing to overlook all the little ways it never quite works. For example, no circle we draw is ever exactly perfect, so in the real world, the Euclidean law of circumference (2[pi][r]) will always be wrong, by some tiny amount. The choice to overlook this law's failure is a human choice, not one the universe makes. The universe is quite content with wonky circles.

That a system conforming to Euclid's axioms will also conform to Euclidean conclusions is a product of the fact that the conclusions are already inherent in the axioms. That humans have to engage tremendous labor to discover these consequences of those axioms is another example of human limitations, but since these consequences follow from those axioms in all possible universes, even universes that have nothing anthropocentric about them, the success of Euclidean geometry has nothing to do with the universe being
anthropocentric. Instead, it has everything to do with our willingness to use such an imperfect tool to describe and predict a messy world, and even then this tool only works well enough when some part of the world just happens to almost conform to Euclidean axioms. When it doesn't, we try something else, whatever we find that happens to work. Hence if nothing ever conformed to Euclid's axioms, we would instead be talking about a geometry based on some other set of axioms, whichever set did occasionally conform to the world, at least near enough to be useful. Since every possible universe will have some geometry that describes it, it's just silly to act surprised when one does.

Newton's laws operate the same way. Like Euclid, Newton began with axioms. The most fundamental of these are more correctly called Newton's Laws of Motion, which were not mathematical formulas, but hypotheses stated in plain English (or Latin, as the case may be: for how they are stated in English see Newton's Axioms of Motion). Newton then argued that if these three axioms held (in conjunction with certain other conditions on a case-by-case basis), then certain consequences followed regarding the motion of objects in the universe, and this is where all his mathematical formulas come from. What is generally overlooked is that, unlike the conclusions of Euclid's geometry, Newton's three axioms don't suffice to generate any of the mathematical equations that are sometimes referred to as Newton's laws of motion and gravity. Those equations only follow when a huge number of additional assumptions are introduced, which have the deliberate effect of keeping the math simple. Those additional assumptions amount to hidden axioms, and these, like Euclid's axioms, never perfectly describe anything in the real world, and frequently don't even come close. Thus, reality is not making Newton's formulas "beautifully simple." We are.

If we chose to, we could build immensely complex (and thus hideously ugly) formulas describing the motion of objects, using the same three fundamental axioms, by incorporating all the incidental factors that change from moment to moment. And yet those ugly laws would be more accurate than all the familiar Newtonian formulas everyone finds so pretty. For example, we could add air pressure to the equations. We could add elements pertaining to the position and velocity of the moon and sun. We could add to Newton's laws variables pertaining to magma displacement in the earth's core, the absorption and evaporation properties of falling bodies, and whether KCET is broadcasting today and how far we are from its transmitters. But we choose not to.

Why? Because the simplest equations are good enough for most human needs. But the universe didn't choose that. It clearly prefers the reverse. Contrary to Steiner and Howell, the universe did not anthropocentrically choose the simple and "beautiful" Newtonian equations of motion. Rather than choosing to obey the simple equations, the universe chose to have bodies always falling according to the most complicated and ugly Newtonian equations imaginable. In fact, apples fall according to mathematical formulas fully beyond any human ability to discover much less work out and employ. But by sticking with the simplest equation, we get results "good enough" for us. And still, only in some cases. Sometimes we need messier equations, but even then we never end up with an equation that exactly describes what will happen. We always choose the simplest equation we can get away with. That has nothing to do with the universe being anthropocentric. It has only to do with humans being practical. Humans thus chose to break down the complex world into simple component behaviors, to make it easier on us. But the universe couldn't care less.

Thus it seems to me that Howell ignores the real world.
When he asks whether my "background in history" makes it "difficult for people [like me] who were not trained in science to appreciate how absolutely uncanny is the continued use of mathematical formalisms by physicists," I see things quite the reverse: Howell's armchair obsession with numbers and formulas seems to have taken him so far from reality that he can no longer see how messy the real world actually is. He fails to see how we are the ones who broke everything down into its simplest components, so we could try to get a handle on complex phenomena by finding their simplest roots. Maxwell was likewise looking for the simplest elements underlying the behavior of electrical systems, and luckily stumbled on some. Not surprisingly, either, as it is inherently probable (especially in a mindless, unplanned universe) that any complex event will be composed of a large array of simpler ones, and that a repeating phenomenon will more often be the result of the repeating of fewer causes than the repeating of several.Hence there is nothing about Maxwell's discovery of electromagnetic radiation that "is precisely analogous to using the number ten as a means of unlocking secrets to the universe" (Howell does not argue that there is anything special about the number ten, he means only that such a thing would be anthropocentric if that were true). Simplicity of component parts just isn't the sort of thing that is unexpected on naturalism, nor is the human tendency to test simpler theories first, or to look for the simplest repeating causes.
 
In Sum

What the likes of Steiner and Howell fail to recognize is that there is nothing uncanny about the success of languages. We invented them to be successful. Mathematics is simply more precise than (for example) English, and thus scientists prefer mathematics as a language because they prefer being precise. That is a human choice. The only way the universe could thwart us in that choice is if it stopped behaving consistently, so that precision of description would no longer produce any benefit. Yet if there were any consistency at all, then more precise descriptions would still be more successful than less precise descriptions, and thus even in such a bizarro universe mathematics would work better than ordinary languages, and thus scientists would still prefer the math. Howell can only get us to a universe that couldn't be more usefully described mathematically by making that universe so fundamentally devoid of consistent behavior as to make life impossible there.

Sure, we can then ask why our universe behaves so consistently, but once we do that we are no longer talking about mathematics or beauty or even anthropocentrism. For merely being consistent is neither uniquely mathematical nor always beautiful, and such a thing does not entail any concern for man. Proposing a rule like "consistency only results from someone's desire" would simply beg the question against any evidence to the contrary. The consistency of the universe can logically have any number of explanations, and naturalists have many plausible hypotheses on the table, none of which are any more bizarre than "God did it" (a God, we must suppose, who is himself inexplicably consistent). And true, we can also ask why humans get so emotional about simple or economic mathematical equations, but I already have enough to say about that in my article on Steiner (
though for more on why humans find simplicity beautiful, and why this is useful, check the index in my book Sense and Goodness without God for "beauty" and "simplicity").

I've said enough on Howell's abuse of my article. I suppose I should also chastise Howell for his arrogant assumption that I have no training in science. I happen to be a historian of science (evidently something he had not imagined). I'm a long-standing member of the History of Science Society and am completing my dissertation in ancient science. I have also formally studied mathematics all the way to the level of calculus. I have a semester of college credits in electronics engineering from the Navy, and worked in the field of electronics and sonar for the Coast Guard, studying for the latter the physics of sound. I also took a seminar in experimental laser physics, and majored in the sciences in high school (all my electives were science courses, receiving perfect grades in physiology, ecology, physics, biology, and other subjects), and in college I took (and received top grades in) several university-level courses in the sciences (including cultural and physical anthropology, theoretical and laboratory geology, and statistical mathematics). I have even published on a scientific subject in a peer reviewed journal (involving, incidentally, mathematics).

But such idle presumption doesn't bother me. Perhaps he would be happy to qualify his psychological hypothesis in just such a way as would exclude me from those who have what he wants to call "training in science," and then still claim that I fail to appreciate how successful mathematical descriptions of the universe have been. Such a conveniently qualified hypothesis would still be falsified by the fact that I clearly do appreciate that. I just understand it better than Howell seems to. My real-world experience may have something to do with that, though I don't presume to know. All I can say is that Howell sure sounds like someone whose ass is firmly in the armchair. Real scientists, for example, work out the kinds of observations that would falsify their hypothesis (in Howell's case, what the universe would actually look like if he was wrong), and Howell does not seem to have done this.
Be that as it may, Howell's paper looks like the work of a hack to me. Deliberately ignoring all arguments and evidence contrary to your view in the very article you claim to be addressing, misrepresenting your opponents (especially by altering your quotations of them), conflating their arguments into a straw man, attacking irrelevancies while avoiding the most substantial points, and then pretending you've attacked something pertinent to your thesis, are all the marks of hack philosophy. But let anyone compare my article with his and judge for themselves.

11 comments:

DFB said...

So, let me get this straight. Some humans designed a language in order to more accurately describe the universe, and then other humans are in awe of the fact that this language describes the universe so accurately. Okay. Got it. It seems similar to marvelling at how wondrous it is that my new suit fits me so well after it comes back from the tailor.

I appreciate you pointing out how hopelessly imprecise our equations really are.

Pikemann Urge said...

I need to understand something, Richard, if you don't mind going over some ground once again. My question is simply: are physical laws fundamentally pure?

Now, forget the real world. Let's pretend we have our universe with only two bodies, each about the mass of a wrecking ball and made of pure carbon or silicon or whatever doesn't evaporate - nothing else happening. Will those two bodies behave towards each other in a 100% elegant and predictable fashion? Remember we have no radio waves, gravity from Saturn etc. to interfere.

Now replace our massive objects with two independent hydrogen atoms. Will our two hypothetical atoms behave predicatably? I'm guessing not but I don't really know why that would be.

So are the physical equations fundamentally 'pure'? Or would they still be very complicated (i.e. would Newton's equations still not be 100% sufficient) even in a theoretically 'clean' universe?

I note, by the way, that we need complex equations to describe simple things (like the trajectory of an artillery shell) but we can use dead simple ones to produce the infinitely complicated and variable objects we call fractals.

One more thing: the points you raise I think are more important for the general public to understand than the actual science itself. I say this because armed with knowledge like this we are 'immune' to being seduced by Creationism.

Richard Carrier said...

DFB: Well said.

Pikemann Urge: My question is simply: are physical laws fundamentally pure?

I don't know what you mean by "pure." Is that a euphemism for "simple"? Broken down into component parts they are simple, but that's simply because we broke them down into simple parts. In practice, all those parts are operating as a complex whole.

Pikemann Urge: Let's pretend we have our universe with only two bodies, each about the mass of a wrecking ball and made of pure carbon or silicon or whatever doesn't evaporate--nothing else happening. Will those two bodies behave towards each other in a 100% elegant and predictable fashion? Remember we have no radio waves, gravity from Saturn etc. to interfere.

The proposed conditions are impossible in the present universe. For example, quantum mechanics entails the intervening space will always be occupied by a sea of randomly changing virtual particles (which will interact with the two large masses to some small degree), while relativity theory entails the forces acting between the two stable bodies will change with their mutual distance from each other, which changes dynamically, causing changes in their relative masses, length of the space in the direction of motion between them, and the rate of time flow between them, all of which add complication upon complication to any Newtonian attempt to describe their motion.

Also, since the two bodies will have different forces acting on different parts of them (basic tidal effect: their ends facing each other will be pulled harder than their opposite ends, etc.), they will undergo expansion and compression (they will be expanded along the axis of motion, which will pull together the polar axes and thus squish the object slightly as it moves), which creates friction, so they will heat up and radiate energy, and thus lose mass (as well as hit each other with some of the consequently radiated photons), and also change shape, which will further change the equation of reaction between them (which will already start far more complicated than any simple Newtonian equation, which only solve for point particles, which don't exist; a proper equation would have to take into account the gravitational effects as they vary across the entire volume of each body, but since those volumes will change in relative shape dynamically as they approach each other, you can see perhaps how impossibly messy the math will be). I might even be overlooking yet other things, but these all amount to quite enough.

Given the above, I would say that even in the imagined conditions, unless you intend completely different laws of physics than obtain in our universe, the equations describing the motion of the two bodies would be enormously complex, and thus neither simple nor pretty. But assuming we could figure out the relevant equations and had good data for the initial conditions, and assuming you count the construction of accurate probability tables for all possible outcomes as "predicting" the behavior of the system (since quantum effects will inject some small randomness to it all), then they would behave in a 100% predictable fashion, but the predicting equations could not plausibly be described as 'elegant' and not at all as 'simple'.

Pikemann Urge: Now replace our massive objects with two independent hydrogen atoms. Will our two hypothetical atoms behave predicatably? I'm guessing not but I don't really know why that would be.

Remember a hydrogen atom is actually a system of several interacting parts (leptons, quarks, and their exchange particles), each with different velocities, masses, volumes, locations, etc. (which in fact constantly change: one of the fundamental findings of quantum particle physics is that the protons in a hydrogen atom, for example, constantly dissolve into their component parts and reassemble, exchanging particles with each other and with orbiting electrons, all within rapid periods of time). And atoms are certainly even more subject to quantum effects than larger scale objects, and thus would never be describable with mere Newtonian equations (in fact, that's one of the fundamental differences between quantum and classical dynamics: individual atoms don't obey Newton's rules very well). So no, nothing very simple here, either.

Pikemann Urge: I note, by the way, that we need complex equations to describe simple things (like the trajectory of an artillery shell) but we can use dead simple ones to produce the infinitely complicated and variable objects we call fractals.

Which raises the question of what counts as simple. A fractal given the same exact input will always produce exactly the same pattern (no matter how complex). Chaos only results from the fact that the input always varies (and even the tiniest variation has radical effects on the outcome).

But this is a whole other problem. In short, fractals as appear in the real world, are simply the effects of repeating processes. The process is simple, but its repetition when interacting with relatively random inputs produces highly complex outcomes, but as a necessary consequence of the geometry of the system. Fractal geometry is thus subject to the same observations as I made for Euclidean geometry.

Blue Devil Knight said...

DFB: one of the best comments I have seen on any blog, anywhere. A wonderful quip.

I am sympathetic to this perspective, but I wonder about the claim that mathematics is 'just another language.' For one, there are a lot of differences between ordinary language and mathematics. The axiomatization of mathematics is one major difference. Also, true claims in mathematics have a kind of unassailability that you don't get in ordinary language. You get it in logic, but that is another formal language and the same applies to that.

Regardless, let's make the reasonable assumption that mathematics and logic were built built to help us make truth-preserving inferences about the world (logic) and precise predictively fecund models of the world (mathematics). Was the axiomatization of the initial patchwork of ideas inevitable? Most importantly, how much flexibility is there in logical and mathematical formalisms? By that I mean, are such formalisms contingent, could we imagine worlds in which logic and mathematics were quite different? If so, that would do severe damage to the Platonist. If not, the Platonist still has a fulcrum about which to get some leverage.

My go-to gal for all this stuff is Gila Sher, especially her Is logic a theory of the obvious? and Logical consequence: an epistemic outlook. Brilliant essays.

Michael said...

The taylor anology is funny...but it's more like your taylor makes a suit designed for ewoks and it turns out to perfectly fit the next ape species we discover in the amazon...

or... you design a universe according to your personal subjective notions of interior design or aesthetics and it turns out to be the one we live in...

I mean, complex numbers were not made to describe anything in the real world, then we found they do (quantum mechanics etc); neither was non-euclidian geometry (then we got special and general relativity)....and set theory describes our interior world...and yet...

Therefore God is either inconsistent or incomplete

OK I'm kidding about set theory and God. And I don't know whether the points above argue against naturalism.

But the language of mathematics has these built in rules of consistency (also present to some extent in other languages) that somehow apply to as-yet-unknown physics even though those rules are supposed to have been generated in our brains... So I think the PREDICTIVE effectiveness of mathematics remains REMARKABLE. I don't know whether that point applies to the Maxwell example under discussion though...

Blue Devil Knight said...

Michael said:
but it's more like your taylor makes a suit designed for ewoks and it turns out to perfectly fit the next ape species we discover in the amazon...

complex numbers were not made to describe anything in the real world, then we found they do (quantum mechanics etc); neither was non-euclidian geometry (then we got special and general relativity)...

But the language of mathematics has these built in rules of consistency (also present to some extent in other languages) that somehow apply to as-yet-unknown physics


These are great points.

ari said...

Simple, complex, how do you count the sample space of ideas?

Agnostics_R_Us said...

RC,

You have no idea how much of this "we have to have an imaginary foundation or reality doesn't exist" crap I get on my site. "The earth doesn't have a foundation! It floats in space! Therefore we can't walk on it!" These theolosophers are effing hysterical about every little misunderstanding and hairsplitting non-problem they can possibly scrape from the bottom of the epistemic barrel. Glad to see I'm not the only one. lol

ARU

Richard Carrier said...

Blue Devil Knight: I am sympathetic to this perspective, but I wonder about the claim that mathematics is 'just another language.' For one, there are a lot of differences between ordinary language and mathematics.

I describe the nature of those differences in my book Sense and Goodness without God (see "mathematics" in the index) and in my online articles on Steiner and Reppert (as noted in my blog): component simplicity and the elimination of ambiguity. What defines mathematics as a distinct category of languages is exactly those two features, which we chose because they are useful together, which is why math exists. What you point out (e.g. the ability to axiomatize and, once you accept the axioms, the inability to deny the conclusions that then follow) are simply the inevitable consequences of these two more fundamental features, which amount to two basic choices we made: to eliminate ambiguity and to construct sentences using only the simplest possible terms (e.g. numbers only ever mean one thing, and each number represents almost as simple an entity as we can imagine, in contrast with words like "man" or "mortal" or "Socrates" or even "is," which is more complicated a word than people often seem aware, e.g. "a cat is a feline" and "a cat is a mammal" use "is" in two different ways).


Blue Devil Knight: The axiomatization of mathematics is one major difference. Also, true claims in mathematics have a kind of unassailability that you don't get in ordinary language. You get it in logic, but that is another formal language and the same applies to that.

And what prevents the axiomatization of propositions in other languages? Only the ambiguity and/or complexity of fundamental terms. Thus, once we choose to create a language that eliminates those two features, axiomatization becomes a possibility within that language (a fact Aristotle was aware of and began to describe, and in fact is the reason the Greeks invented mathematics and logic as formal systems to begin with). That's why "logic" also can be axiomatized: because we made it to be.

Ultimately logic, like math, is basically just a computational procedure, and like all procedures, it is a man-made technology. But like other technological procedures (say, growing corn), we can't just make up anything, but only what works, which is limited by the nature of the universe (hence my discussion of the ontology of logic in my article on Reppert).


Blue Devil Knight: Could we imagine worlds in which logic and mathematics were quite different? If so, that would do severe damage to the Platonist. If not, the Platonist still has a fulcrum about which to get some leverage.

I don't think a Platonist would be stuck on this. He could just add the alternatives to his Platonic "megaverse" of ideal forms (whatever that is supposed to mean--Platonists have yet to explain to me in any intelligible way what the hell it is they actually believe). At any rate, I discuss the possibility of imagining other worlds in my book (see "contradiction, nature of" in the index) and in my article on Reppert (where I discuss the ontology of logic).


Michael: I think the PREDICTIVE effectiveness of mathematics remains REMARKABLE.

In any possible universe, some language will do very well at formulating (and thus making) predictions. Let's say in an alternate universe math does lousy (though as I explain in my blog, I suspect life would be impossible in such a universe, set that aside for now), but German does uncannily well. You would then be saying things like "I think the predictive effectiveness of German remains remarkable." Then a guy in another universe where Esperanto does uncannily well will say the same thing about Esperanto, etc. For any of this to be remarkable, it must actually be highly probable that any random universe will not be well described by any language. But that seems highly unlikely to me, given that the possible languages are infinite, hence any random universe will surely be well described by one of them. Thus that we found one such language here is not remarkable at all, especially since we were specifically looking for it.

But I think the matter is even simpler than that, since all math does is describe geometry, which is simply a description of space-time. Therefore every universe with space and time in it will be maximally well described by geometry, hence math. Therefore there is nothing remarkable about a universe with space in it being well described by math: since all "predictions" are nothing more than descriptions of what will happen, which are really nothing more than descriptions of what does happen, which are really nothing more than descriptions of what is, which is really nothing more than a description of the universe itself. So I see nothing remarkable here. To say math does well at predicting things here is to say nothing more than that this universe consists of (at the very least) space and time. I don't see that as a surprising conclusion. We already knew that by looking around, well before we even had language, much less invented math.


Michael: But the language of mathematics has these built in rules of consistency (also present to some extent in other languages) that somehow apply to as-yet-unknown physics.

So does English. Ever read fiction? We can describe countless worlds with different physical laws, using English alone. Any Sci Fi book shelf is full of examples. In the case of math, we are doing exactly the same thing, only with greater precision (by eliminating the complexity and ambiguity of our words). That is, in fact, the only difference. This ability to take parts of an actual world and reorganize them to create fictional worlds is actually how animals (even before we became conscious) learn to anticipate how the world will work in novel circumstances: by experimenting with combinations, some of which will actually be encountered some day. I discuss this in my book, under theory of mind.

This does not mean all math is fiction, but it does mean some math is. Hypothesis-formation is simply the construction of experimental fictions that we then test to see if they describe what is real and not fictional after all. But since math keeps its component terms simple (you won't find any isolated words there that come anywhere near as complicated as "tree" or "go"), it will always describe truthfully any system that conforms to the axioms and terms of any mathematical sentence, and most of the universe thus conforms (in fact, all the universe conforms to the axioms of some geometry, because everything in the universe exists in space and time; and since all number theory follows from geometry, so does arithmetic and every other mathematical system).

By analogy, when Theophrastus described "typical" personalities (which psychologists still do, more scientifically now), he was writing fiction (in Greek, an ordinary language), but the universe actually does contain the types of people he was describing. Math works the same way. 2+2=4 is essentially a fiction (it refers to no particular thing, but a common pattern of adding two things to two things), but like the personalities described by Theophrastus, it describes a commonplace reality (very often two things are in fact added to two things). As long as there are discreet things at all, and any adding of them together, any language that describes the adding of discreet things will describe numerous repeating events of that universe. The only way it could be otherwise is if there was no adding and no discreet entities at all, but if that were the case, we would not exist (in fact, all thought would be physically impossible in that universe).

Similarly, when literary critics talk about a work of fiction being "unrealistic" they are responding to exactly this feature of fiction: the particulars might not exist, or their arrangement together might not exist, but the generalizations that describe what would then happen (if, for example, you put a particular type of person in a particular type of situation) must conform to reality for fiction to be convincing (unless your aim is, say, comedy). Thus, ordinary English can correctly anticipate situations and outcomes never yet experienced, just as math can anticipate "laws of physics" not yet discovered.

So, again, I find nothing remarkable here. To the contrary, I find exactly what I should expect if naturalism were true.

I should add that all this (the ability to predict well the expected outcome even with ordinary English) follows even for completely alien worlds (in which ghosts exist or The Force or sorcery, etc.). And it is possible (even if now highly improbable) that we actually do live in one of those alien worlds. So, too, for physics, it's just that--unlike authors of good fiction--scientists don't spend much time talking about mathematically possible physical laws that are empirically improbable. But they could if they wanted to.

Robert O'Brien said...

I am familiar with Dr. Howell's article, and I agree with him, Aristotle redivivus. There are many things in mathematics that need not be so clean, such as Cauchy Integral Theorem from complex analysis.

I am not impressed by the scientific "credentials" you offer, and given your intoxication with Jesus Mythicism, I am not even impressed by your relevant credentials.

Richard Carrier said...

Robert O'Brien:

I'm sorry, but...do you actually have an argument?