Sean Carroll has an interesting piece at The Stone on the New York Times site, pointing out that the theory of cosmic inflation was motivated by naturalism. In other words, it was motivated by the desire to find a natural explanation for something that didn’t look natural, such as the apparent fine tuning necessary for the universe to be flat, among other things.

But it reminds me that science and naturalism go hand in hand. The first known scientists, the pre-socratic philosophers beginning with Thales, were motivated to find natural explanations for phenomena. Their methods wouldn’t pass muster as science today, and many of their ideas were wildly off the mark (although Thales was able to predict lunar eclipses), but it was the approach of looking for natural explanations that set them apart from their predecessors, who had generally invoked gods or spirits as explanations.

As Carroll points out in his article, the assumption that there must be a natural explanation has been an enormously productive one throughout the history of science. Any attempt to work outside of this assumption has generally failed to provide the reliable knowledge that science is known for.

It’s one of the reasons teleology, trying to figure out what everything’s purpose is in the universe, was abandoned by scientists centuries ago. At the beginning of the scientific revolution, philosophers often insisted that it wasn’t enough to understand how a phenomena behaved or how it came about. One had to understand what its purpose was in the overall scheme of things.

As science came into its own, it quickly became apparent that teleology wasn’t a productive endeavor. There simply wasn’t any way to test ideas of what those purposes might be, to acquire reliable knowledge of them.

Does this mean that science is inherently non-theistic? I think it is. Not in a manner that denies the existence of gods, but in a manner that simply fails to find them, or any other supernatural concept, useful as explanations. (Although there are definitely conceptions of gods, some would say naive ones, whose existence is contradicted by scientific evidence.) This isn’t philosophical naturalism, but methodological naturalism, the working assumption that only looking for natural explanations is productive.

This raises the issue of what is natural, as opposed to what is supernatural? And that turns out to be tricky to nail down if you think about it. If the natural is what occurs in nature, that is following the laws of nature, then the supernatural must be what occurs outside of the laws of nature. Of course, if we don’t understand the laws of electricity, then lightening looks supernatural (as it once did).

Ultimately though, the supernatural is what we hear about in stories that are not consistent with the observed laws of nature. It is violations of those laws that occur in an untestable manner. For this reason, it’s often stated that the supernatural is outside the scope of science.

As a skeptic, I don’t think the supernatural exists, but I have to admit I can’t prove that it doesn’t. Of course, I also can’t prove that on a certain date in a certain location, 1+1=3 wasn’t true for a five minute period. Such an occurrence is outside the scope of mathematics, it is supermathematical, but that doesn’t mean I’m going to seriously consider that it might have happened.

But this is a philosophical position I hold, not a scientific one. Scientifically we can say that lightening strikes follow the laws of electricity, and that there is no evidence of them targeting evil doers. But scientifically we can’t say that there is no force that targets evil doers in some manner, only that there’s no evidence for it.

So, strictly speaking, science doesn’t demand philosophical naturalism, the idea that nature is all there is. But it does limit its scope to naturalistic explanations. And that has been a huge component of its success.

###### Related articles

- Unity of Science (theleatherlibraryblog.com)
- Ionian/Eleatic Philosophy and the importance of Logos in the Polis- The Beggining of Greek Philosophy. (eggflip.wordpress.com)
- Unexplained terminology Explained: ‘Paranormal’ versus ‘Supernatural’ (idoubtit.wordpress.com)
- Towards a Naturalized Ontology: How to Arrive at a Naturalized Ontology (cameronafzal.wordpress.com)

Hi SAP,

“If the natural is what occurs in nature,”

I think a more rigorous distinction between natural and supernatural is required.

If ghosts exist, then ghosts exist in nature, and so ghosts are by definition natural. And so are miracles, and angels, and psychic powers, etc. If an intervening God exists, then the laws of physics are not broken but amended with an asterisk beside each one referring to the footnote: “unless God wills otherwise”. What happens still obeys physical law.

I propose instead that naturalism is about mathematics. All natural phenomena must ultimately be reducible to laws which can be represented entirely mathematically.

http://disagreeableme.blogspot.co.uk/2012/06/super-naturalism.html

LikeLike

Hi DM,

An interesting comment and post. I’d agree that if ghosts and gods exist, then they are natural, and we’d have to find a way to understand them. Though I’d be reluctant to pre-conclude that they couldn’t be understood mathematically. The hazy demarcation was what I was trying to get at with my lightening example.

On naturalism being about mathematics, I can definitely see it under the MUH (mathematical universe hypothesis). I’m still making my way through Tegmark’s book (currently in chapter 6, the chapter on level I and II multiverses), so I’m still not sure how I feel about the MUH.

I think one hangup I still have, that I’m hoping Tegmark addresses, is the simple brute fact of existence versus non-existence in this universe. E=mc

^{2}is mathematics, but so is E=mc^{-4}. One exists, the other doesn’t. I know energy mass equivalence was worked out mathematically as a consequence of special relativity, but special relativity couldn’t be worked out mathematically from Newtonian mechanics. Empirical observations that violated NM were necessary. If the only answer to this question is multiverses, I don’t think I’m going to find it satisfying.I’m also wondering what effect the existence of irrational numbers, like pi, have on this idea. It seems like they are cases of mathematics not being able to precisely model reality.

LikeLike

I’m not pre-concluding that ghosts can’t be understood mathematically. In fact if they existed I’m certain they could! But those of a supernatural bent don’t really seem interested in working out the details of how these supposedly real phenomena work, or if they do they suppose the details follow narrative logic rather than mathematical logic.

The idea of the Abrahamic God is a perfect example. God is supposed to be perfectly intelligent, omniscient etc, but if all that is natural is mathematical then a natural God’s mind would be an algorithm just like any other and no algorithm can be perfect or omniscient due to limitations such as the halting problem, Godel theorems etc. Attributing omniscience to God is supernatural because it is supermathematical.

I doubt Tegmark will satisfy you, but I think he’s right regardless (although there are ways in which I would make the argument differently). Looking forward to your thoughts when you’ve finished the book.

LikeLike

Good points.

I’m definitely going to do a post when I finish the book. I’m enjoying it. I can say right now that I’m having the same reaction as you did to Level I multiverses. Sure they exist, but calling them “parallel universes” seems slippery.

LikeLike

The belief in the supernatural presupposes that there is some connection between the supernatural and the natural in order for an effect to traverse the divide (otherwise, there would be no cause for belief in the supernatural, because nothing would happen), which leaves you with the chore of deciding where the dividing line between the two is. The line was once entirely to the right, with all things described supernaturally. The line is now almost entirely to the left, with supernaturalism so well punished that people have to make up new things to believe are beyond this reality, or alternatively, pursuing ignorance to retain the last refuges of the supernatural.

It doesn’t take mental gymnastics for me to conclude that the trend against the supernatural will continue, and personally, I feel that it has gone far enough to assume that there are no agendas being pursued by great, cosmic entities. At least not in our slice of the universe.

LikeLike

I like your qualifier at the end.

LikeLike

I think mathematicians can be sure that 1 + 1 was never equal to 3 at any place or time, because mathematical proofs do not depend on exterior influences. They are logically self-consistent and universal.

“I’m also wondering what effect the existence of irrational numbers, like pi, have on this idea. It seems like they are cases of mathematics not being able to precisely model reality.”

Think about why pi has the strange value that it does. Start with a square as a first approximation of a circle. If the radius of the circle is 1, then the area of the square that encloses it is 4. We can estimate pi as 4. To get a better estimate of the value of pi, draw another square at 45 degrees to the first. The area enclosed by the intersection of the two squares is 3.5, giving an improved estimate of pi. The more squares we add, the closer we get, but to find the true value of pi we need an infinite number of squares, so pi is the sum of an infinite series. That is why it has an irrational value. So in a way I think you are right that the circle is an idealised form that can be modelled arbitrarily close, but never reached.

LikeLike

Of course it makes sense that Pi is irrational, but I don’t think your explanation quite works. There are infinite series which converge to rational numbers after all.

LikeLike

DM, I’m not quite sure I see how pi being irrational make sense. It’s always seemed a bit bizarre to me that we can’t nail down the ratio between a circle’s circumference and diameter.

LikeLike

Hi SAP,

I didn’t get into that argument because I was replying to Steve, with whom I agree about Pi being irrational. I’m saving the argument with you until you finish Tegmark’s book.

But I can’t ignore this any longer. I don’t see how an irrational Pi doesn’t make sense. It’s no more surprising than that the square root of two is irrational. And I would say that we can nail down the ratio, it’s just that it cannot be expressed as a ratio between two whole numbers. I don’t see why we ought to expect it to be rational and I certainly see no reason to construe this as mathematics failing to model reality precisely. Quite the contrary!

Granted, at first blush it may be surprising. It was surprising to me when I learned about it as a ten year old. But just because mathematics is more complex than a naive person might expect is no reason to think that there’s something wrong with it.

LikeLike

Oh, I’m sure we’ll have plenty to discuss when I finish the book 🙂

“And I would say that we can nail down the ratio, it’s just that it cannot be expressed as a ratio between two whole numbers.”

So, as I’ve mentioned before, mathematics isn’t my strong suit, so forgive me if I’m asking a silly question. But what other way is there to express a ratio but as a relationship between two numbers? Isn’t that fundamentally what a ratio is?

LikeLike

A ratio is a relationship between two numbers. Those numbers do not have to be whole. We know that the ratio of the diameter of a circle to its circumference as the ratio 1:Pi. Pi is perfectly well defined, so this is pinned down in my book.

If you think that Pi is not well defined because there is no way to write down its digits precisely, I think that overlooks the fact that there are pretty simple infinite series that equal Pi, for example:

LikeLike

Interesting. That almost seems like a restatement of Steve’s point above. Not sure how having an infinite series improves the situation though.

I see our inability to define pi without just dropping the symbol in its place as an issue. It’s not a pragmatic one of course, since we can know it well within any practical measurement inaccuracies, but it seems like a theoretical one for the MUH. That said, I’ll freely admit it might be due to my mathematical naiveté.

LikeLike

The point I make is similar to Steve’s, but there is a difference. Steve says (paraphrasing) “Of course it is irrational because it is an infinite series”, but that’s not quite right because there are infinite series which are not irrational. Nevertheless some irrational numbers can be defined as infinite series, and Pi is one of them.

“I see our inability to define pi”

We have no such inability. The infinite series is a definition of Pi. There are many other such definitions and they are all equivalent.

LikeLike

Footnote: I didn’t say that pi is irrational because it is a sum of an infinite series. I was saying that because it’s the sum of an infinite series, we shouldn’t be surprised that it is irrational. If you add an infinite number of fractions, then the sum will be irrational more often than not.

LikeLike

Fair enough, Steve, that’s a good clarification.

LikeLike

I think your first paragraph gets to the debate on whether or not mathematics ultimately rest on empirical foundations. Certainly, once you get above those foundations, the conclusions aren’t constrained by physical reality but by logical necessity. However, I think if we saw a single stick, added a second stick, and regularly observed that we now had three sticks, mathematics would be effected.

LikeLike

I finally tore myself away from ‘Is science more “unnatural” than religion?’ to read this post. Thanks for the “pre-socratic philosophers”, “Thales”, and “philosophical naturalism” links. There’s hope yet that someday I’ll be able to say that I know something of philosophy 🙂

LikeLike

My pleasure. Given your talent for research, I suspect you’ll be teaching me things about them soon 🙂

LikeLike

Hmm? WordPress didn’t tell me you replied.

I don’t think it too likely I’ll be teaching anyone much about philosophy in my lifetime beyond ‘Change is the only constant’. I’ll be quite happy if someday I can nod my head and say “Uh-uh” and “I see” appropriately and convincingly when the topic comes up in conversation. My preferred philosophies tend to align with:

http://imgfave.com/view/1092441

… and no, *not* the biped on the right 🙂

LikeLike

LOLS! The thought of the biped on the right is incomplete. The full version is, “What does it all mean, in order that I might better eat, survive, and reproduce?”

LikeLike

Ha! I’ve never seen the complete version – even better!

LikeLike

science is inherently non-theistic

sure

that is true… debates of science vs religion are typical of classic task avoidant behaviors

http://scholar.google.com/scholar?q=task+avoidance+behaviors&hl=en&as_sdt=0&as_vis=1&oi=scholart&sa=X&ei=MlI3U9bFMYrMqAHe24HYCw&ved=0CCcQgQMwAA

LikeLike

Well, I’m definitely a chronic procrastinator, but what task, or tasks, would you say are being avoided?

LikeLike

Scientists avoid doing science and religious avoid getting spiritual… people always avoid tasks… you aren’t alone Mike.

LikeLike

🙂

LikeLike

You must have sympathy with someone like Richard Dawkins who has devoted much of his professional life to educating the public about evolutionary biology, only to discover that half of Americans don’t believe in evolution because of a literal reading of the Book of Genesis. That must be more than a little frustrating. Dawkins clearly sees that his task as not simply explaining biology, but in engaging with religious fundamentalism.

LikeLike

Dawkins can care for himself. I posted one of his debates.

He wasn’t seemingly interested in fundamentalist when he was debating Chopra. Deepak certainly is not a fundamentalist Christian. He’s rich and famous. Why would I worry about him?

LikeLike

Eric, I wasn’t suggesting you should worry about Dawkins. I’m saying that if religious fundamentalists tell a scientist that his life’s work is a waste of time, then it is rational for him to go on the offensive. He isn’t just procrastinating.

LikeLike

Do you conclude that Dawkins ought to feel threatened by a fundamentalist view of evolution – one that exactly matches Genesis?

I think he can be true to his purpose without arguing that point to a small fraction of religious believers.

If he is unable to remain true to himself, to his purpose, the Ethologist in him in light of criticism about his views on gene selection, memetics, and sociobiology as being excessively reductionist; the fundamentalist Christians views on God as Creator are not his opposite. He thereby is not debating for scientific purposes but to defend a general world-view. For instance, Mary Midgley perhaps rightly points out — science has become religion — perhaps Dawkins treasures his atheistic view as though it is a faith that must be believed until the proof may be found. Dawkins’s meme is a theoretical extension of his arguments about gene selection… not at all “proof” that was in labs replicates as scientific law. His work as an Ethologist was placed on the back burner since that time. He emerged not as a scientist any more. His purpose is reduced like his science to debate himself into the work of defining a world-view.

LikeLike

I think he largely sees his purpose as the public understanding of science. Since half of Americans reject his science, he feels that he has a lot of work to do in this regard.

LikeLike

perhaps then his critics are right — perhaps science is of no great importance to him and he prefers to debate philosophy

LikeLike

I wish he would debate philosophy. I wasn’t aware he had any understanding or interest in it.

LikeLike

LOL

LikeLike

Very interesting post and comments. I would agree with @Disagreeableme that if a phenomenon exists then it must be natural, and this would include God and his works. It would be interesting to see how an argument might be made that God is not a natural phenomenon.

The problem for physics is that this discussion is almost entirely beyond its boundaries. It is only when we examine the foundations of physics or of the phenomena it studies that these issues arise, and this is metaphysics. Any definition of ‘natural’ must come from metaphysics, I would say, and so far I haven’t seen a good one.

LikeLike

You are right, guymax. The boundaries of physics are of course its most interesting aspect, and always a popular topic for speculation.

LikeLike

Hmm. Not for many physicists apparently.

LikeLike

Right. I shall risk talking nonsense for a minute, because I’m also intrigued by Pi, and perhaps see it as more significant than it actually is. .

I tend to see Pi as a sign that there is something amiss with our ideas about geometry. If r = 1 then we can only find an approximation for c, and this seems to me to suggest an irrational relationship between r and c, which seems to suggest something amiss with our idealised notions of circles and radii. Is this idiotic? Probably.

LikeLike

I would say it is not true that we can only find an approximation for c. We can only represent it approximately if we confine ourselves to a system of representation based on rational numbers, but we can represent it precisely if we use a system of representation based on infinite series.

LikeLike

Perhaps you’re right, but it seems like cheating to me.

I have a friend (ho ho) who wonders if Pi represented the increasing precision of the calculation as the width of the lines of r and c approached zero. The result is bound to be an approximation where we use a thick pencil (conceptually) to draw r and c, since there is no precise point of measurement. Iow, when we measure r, the length is different at the inside edge of c than it is at the outside edge, So we have to make c infinitely thin for a measurement of r, in which case c isn’t there any more. Or something like that. Is this just daft? I expect it is just irrelevant.

LikeLike

What you’re saying may have merit as some kind of poetic analogy or metaphor, but is not strictly speaking true. Not all circles are drawn with pencils, after all. Any physical event has effects that propagate outwards in a sphere at the speed of light. This is a perfect sphere with no border, only a distinction between what is inside and what is outside the sphere. The sphere has a radius and a surface area which can be calculated with Pi.

(At least this is true classically. I guess quantum fluctuations may imply some fuzziness to the border which I guess you could use to rescue your argument. In that case, while I can’t necessarily refute it so easily, I still don’t buy it!)

But Pi also shows up in all kinds of physical and mathematical contexts that have nothing to do with circles or spheres per se.

I think the truth is that there is no elegant way of “rationalising” Pi’s irrationality. It’s just irrational, plain and simple, and that’s only confusing or problematic if you are working from the unfounded assumption that all useful or meaningful numeric quantities should be rational.

LikeLike

Thanks for not laughing!

I’m okay with the meaningfulness and usefulness of such numbers, but they seem to be telling us something that I can’t quite grasp. Your propagating sphere, where everything is inside or outside, seems to bring up Dedekind’s problem, that there has to be a position that is not inside or outside the sphere in order to separate the inside from the outside. Still thinking about this though. It’s definitely relevant.

LikeLike

I’m not sure Dedekind’s problem makes sense, though. If there does have to be such a point, then depending on how we define the sphere such a point on the surface of the sphere either lies within the set of points influenced by the event or outside it.

For example, if we want to classify all numbers as either positive or non-positive, we define positive numbers to be all numbers greater than zero. Zero itself is the border point and is unambiguously not among the positive numbers.

LikeLike

Fair enough. But it’s not among the negative numbers either. So on your view there would be three classes of numbers. This idea seems unparadoxical. I thought that for Dedekind there would only be two classes, one on either side of the cut. This idea is paradoxical.

I hope you don’t mind me exploring this a bit. I’d like to double-check with you that my thoughts are not daft.

As it happens I’m just trying to write something about Hermann Weyl, of whom I’m a fan, and to outline his views on the continuum. He considers Dedekind’s Cut to be an coherent idea in relation to the continuum in the context of mathematics, since we can define the number line how we like, but he suggests that it would be a mistake to assume that this is more than just a useful idea and then go on to turn it into a metaphysical theory of the continuum. In reality not all points can fall on one side or the other of the cut without denying the definition of the continuum. A true continuum cannot be cut since it has no parts. Perhaps it is because he is a philosopher as well as a physicist and mathematician that he seems able to disentangle these issues in a way that few others can.

As I understand Weyl, (which is certainly not completely), he proposes that physics and metaphysics make a mistake when they equate the continuum of space-time, intuition and experience, which is a true continuum, with that of mathematics and current physics, which, as a series of points and moments, is a creation of the intellect with no empirical foundation. This would be my view also for what it’s worth. This mistake, if it is a mistake, would lead to anomalies and contradictions, and for me Russell’s Paradox would be one of them, and all other problems of metaphysics as well.

I sometimes wonder if the endlessness of Pi is a symptom of the same problem. When we assume an infinitely-divisible and yet (nevertheless) extended number line, and a space-time to match, as we must for geometry, then the ratio between the circumference and radius of a circle is infinitely divisible. This does not look like a problem to any mathematicians that I know of but still, I wonder whether it tells us that there is something wrong with our usual notion of the continuum, just as Weyl suggests, or at least is connected to the same issue.

LikeLike

I deliberately didn’t bring negative numbers into it because I wanted to show a partition into two classes. In the partition I proposed, the two classes are positive and non-positive. If you do want to further partition the non-positive numbers into negative and zero, then we have three partitions. But we don’t have to concern ourselves with that second partition if we don’t want to. Dedekind’s cut works just like this.

“they equate the continuum of space-time, intuition and experience, which is a true continuum, with that of mathematics and current physics, which, as a series of points and moments, is a creation of the intellect with no empirical foundation.”

I would say you have this backward. The continuum of mathematics is a true continuum. It’s not a series of discrete points, it’s an infinite set of points, including those defined by irrational (and potentially complex) numbers. The continuum of physics is likely not continuous at all, being possibly discrete at the Planck scale.

I would also endorse mathematical Platonism, which holds that no mathematical object is a creation of the intellect but a discovery which exists independently, and for which no empirical foundation is required.

There are enough genuine problems and paradoxes with mathematics (did you know that the sum of all positive integers is -1/12?) not to warrant much concern over Pi. Irrational numbers are just like zero, negative numbers, complex numbers, etc: kinds of numbers that were once considered not to exist, then accepted as conveniences and finally recognised as full-blooded numbers in their own right. I would hold that any concept of a number which is well defined exists just as much as any other, and so find myself completely untroubled by Pi.

LikeLike

Thanks for clarifying something for me here DM. I’ve just learnt something important that I need to think about before replying. I’ll just say that your cut would make sense where Dedekind’s, according to Weyl, would not. He discusses this in relation to time, where ‘past’ and ‘future’ imply also a ‘now’. Fingers crossed I’m not misreading him.

I’m not suggesting that there is a problem in mathematics, only that problems arise when we try to reify all these mathematical entities in cosmology, metaphysics and the foundations of analysis. I have long believed this to be true, on philosophical grounds, but Weyl i sone of the few mathematician I’ve come across who can make the case in a quite formal way, yet in a way that even I can understand.

I’m also untroubled by Pi. by the way, I just wonder about it. I feel it is telling us something important but can’t see what it is.

Generally I seem to share Weyl’s view on both mathematics and philosophy, but I wouldn’t be able to defend his view of mathematics without making him look incompetent. I do prefer his view, however. sorry about that. I’ll be trying to explain why in an upcoming post about Weyl. . . .

It is two different views of the continuum that divides philosophy into ‘eastern’ and ‘western’, so it is not a small issue.

LikeLike

Sorry SAP. I seem to have accidently posted about Pi in the wrong comments section. Don’t know how that happened.

LikeLike

No worries at all guymax. I do it all the time.

LikeLike

…and I very much appreciate the conversation you guys are having.

LikeLike