Beginning with Hugh Ross, I undertook to critique various articles on the finetuning of the universe for intelligent life that I deemed to be woeful, or at least in need of correction. A list of previous critiques can be found here. I generally looked for published work, as correcting every blog post, forum or YouTube comment is a sure road to insanity. I was looking to maximise prestige of publication, “magic bullet” aspirations and wrongness about finetuning. I may have a new record holder.
It’s an article published in the prestigious British Journal for the Philosophy of Science by a professor of philosophy who has written books like “Introduction to the Philosophy of Science”. It claims to expose the “philosophical naivete and mathematical sloppiness on the part of the astrophysicists who are smitten with [finetuning]”. The numbers, we are told, have been “doctored” by a practice that is “shrewdly selfadvantageous to the point of being seriously misleading” in support of a “slicklypackaged argument” with an “ulterior theological agenda”. The situation is serious, as [cue dramatic music] … “the fudging is insidious”. (Take a moment to imagine the Emperor from Star Wars saying that phrase. I’ll wait.)
It will be my task this post to demonstrate that the article “The Revenge of Pythagoras: How a Mathematical Sharp Practice Undermines the Contemporary Design Argument in Astrophysical Cosmology” (hereafter TROP, available here) by Robert Klee does not understand the first thing about the finetuning of the universe for intelligent life – its definition. Once a simple distinction is made regarding the role that Order of Magnitude (OoM) calculations play in finetuning arguments, the article will be seen to be utterly irrelevant to the topic it claims to address.
Note well: Klee’s ultimate target is the design argument for the existence of God. In critiquing Klee, I am not attempting to defend that argument. I’m interested in the science, and Klee gets the science wrong.
Warning Signs
Klee, a philosopher with one refereed publication related to physics (the one in question), is about to accuse the following physicists of a rather basic mathematical error: Arthur Eddington, Paul Dirac, Hermann Weyl, Robert Dicke, Brandon Carter, Hermann Bondi, Bernard Carr, Martin Rees, Paul Davies, John Barrow, Frank Tipler^{1}, Alan Lightman, William H. Press and Fred Hoyle. Even John Wheeler doesn’t escape Klee’s critical eye. That is quite a roll call. Eddington, Dirac, Weyl, Bondi, Rees, Hoyle and Wheeler are amongst the greatest scientists of the 20th century. The rest have had distinguished careers in their respective fields. They are not all astrophysicists, incidentally.
That fact should put us on edge when reading Klee’s article. He may, of course, be correct. But he is a philosopher up against something of a physicist dream team.
Klee’s Claim
The main claim of TROP is that finetuning is “infected with a mathematically sharp practice: the concepts of two numbers being of the same order of magnitude, and of being within an order of each other, have been stretched from their proper meanings so as to doctor the numbers”. The centrepiece of TROP is an examination of the calculations of Carr and Rees (1979, hereafter CR79) – “[this] is a foundational document in the area, and if the sharp practice infests this paper, then we have uncovered it right where it could be expected to have the most harmful influence”.
CR79 derives OoM equations for the levels of physical structure in the universe, from the Planck scale to nuclei to atoms to humans to planets to stars to galaxies to the whole universe. They claim that just a few physical constants determine all of these scales, to within an order of magnitude. Table 1 of TROP shows a comparison of CR79’s calculations to the “Actual Value”.
Klee notes that only 8 of the 14 cases fall within a factor of 10. Hence “42.8%” of these cases are “more than 1 order magnitude off from exact precision”. The mean of all the accuracies is “19.23328, over 1 order of magnitude to the high side”. Klee concludes that “[t]hese statistical facts reveal the exaggerated nature of the claim that the formulae Carr and Rees devise determine ‘to an order of magnitude’ the mass and length scales of every kind of stable material system in the universe”. Further examples are gleaned from Paul Davies’ 1982 book “The Accidental Universe”, and his “rudimentary” attempt to justify “the sharp practice” as useful approximations is dismissed as ignoring the fact that these numbers are still “off from exact precision – exact fine tuning”.
And there it is …
I’ll catalogue some of Klee’s mathematical, physical and astrophysical blunders in a later section, but first let me make good on my promise from the introduction – to demonstrate that this paper doesn’t understand the definition of finetuning. The misunderstanding is found throughout the paper, but is most clearly seen in the passage I quoted above:
[Davies’] attempted justification [of an order of magnitude calculation] fails. 10^2 is still a factor of 100 off from exact precision – exact finetuning – no matter how small a fraction of some other number it may be [emphasis added].
Klee thinks that finetuning refers to the precision of these OoM calculations: “exact precision” = “exact finetuning”. Klee thinks that, by pointing about that these OoM approximations are not exact and sometimes off by more than a factor of 10, he has shown that the universe is not as finetuned as those “astrophysicists” claim.
Wrong. Totally wrong.
The claim that the universe is finetuned for life can be formulated as follows: in the set of possible physical laws, fundamental parameters, and initial conditions of the universe, the set that permits the evolution of intelligent life is very small. Leonard Susskind puts it like this:
The Laws of Physics … are almost always deadly. In a sense the laws of nature are like East Coast weather: tremendously variable, almost always awful, but on rare occasions, perfectly lovely. … [O]ur own universe is an extraordinary place that appears to be fantastically well designed for our own existence. This specialness is not something that we can attribute to lucky accidents, which is far too unlikely. The apparent coincidences cry out for an explanation.
Lee Smolin says:
Our universe is much more complex than most universes with the same laws but different values of the parameters of those laws. In particular, it has a complex astrophysics, including galaxies and long lived stars, and a complex chemistry, including carbon chemistry. These necessary conditions for life are present in our universe as a consequence of the complexity which is made possible by the special values of the parameters.
To calculate the degree of finetuning of a parameter, one compares the range that permits life to the possible range (often the range over which the theory in which the parameter appears is welldefined or selfconsistent, or the range over which we can make predictions about the resultant universe). More generally, we compare the lifepermitting subset of parameter space to the total possible space. Comparing the results of an approximate calculation with the results of a more exact calculation or measurement in our universe is utterly irrelevant. It fails to understand the definition of finetuning. The degree of finetuning is not the same as the accuracy of certain OoM equations discussed in the finetuning literature. Klee’s paper, even if its mathematical claims were correct, is simply beside the point.
The Role of Order of Magnitude Calculations
Let’s clarify why finetuning papers discuss OoM calculations. To identify a case of finetuning , we need to be able to predict what the universe would be like if the laws, parameters and initial conditions were such and such. This procedure is known as theoretical physics. Focussing on the parameters, we would like a formula that relates the features of our universe to the value of the parameters. The long way is to solve the equations for the laws of nature in all their glory. This is often difficult. As a shortcut, we can often use an approximate calculation to give an equation that has the correct form, but has a different constant out the front.
For example, if one calculates the time it takes for a uniform static sphere of matter (density , radius , mass ) to collapse by solving Newton’s equations, one gets the exact answer:
If we approximate the velocity of collapse by a single velocity, and calculate that speed by comparing kinetic with gravitational energy, we get:
The rough calculation gets the form of the equation correct. The dimensionless number out the front is determined by geometric factors, and often of order unity. In the calculation above, the rough version’s constant is 0.35, while for the exact solution it is 0.54.
Note that such calculations are extremely important in physics. In Cambridge, I tutored a course called “Topics in Astrophysics” which was almost entirely concerned with OoM calculations such as those found in CR79 and standard astrophysics textbooks. Such courses are common. OoM calculations do not replace more exact calculations, but are very important as sanity checks, for building intuition and for understanding the dominant physical processes in a given situation. John Wheeler was famous for saying “never do a calculation without first knowing the answer”. Klee is railing against one of the most important, most familiar tools in the physicist toolkit, not against something invented in the anthropic literature.
CR79, Davies, and Barrow and Tipler (1986) use OoM calculations to illustrate that “there exist invariant properties of the natural world and its elementary components which render inevitable the gross size and structure of almost all its composite objects”. To show that these approximations are approximations does not show that “the mass and length scales for stable systems in the universe are not precisely determined by the four constants [considered by CR79]”. It only shows that these scales are not precisely determined by those particular formulae. The dependence on the constants is correct. The constant out the front is approximate, but can be corrected by more careful calculation if so desired.
Much, perhaps most, modern work on finetuning uses more sophisticated calculations than the OoM calculations of the early papers of Carr and Rees et al. If CR79’s approximate calculations of the mass and size of a star aren’t good enough for you, then repeat the calculation in more detail, as Fred Adams has done. Barr & Khan (2007), Agrawal et al. (1998a,b), Jaffe et al. (2009), Epelbaum et al. (2013) and a hundred other papers (see my review paper for more details and references) use stateoftheart simulations to explore the finetuning of the universe. Klee is wrong when he claims that, if inaccuracy infests CR79, then it infests all finetuning calculations. Even if Klee were correct in complaining about OoM calculations, his complaint is decades out of date.
Why it doesn’t much matter
Consider the following finetuning case, discussed by Robin Collins. If (having fixed a system of units and not used G in doing so) the gravitational constant were increase by a factor of about 3000, stars would burn out within a billion years. The strength of the forces of nature varies up to . Thus, the ratio of the lifepermitting range to the possible range is around 1 in .
This number remains very small even when the various levels of approximation are taken into account. Klee’s claim that that “10^2 is still a factor of 100 off from exact precision – exact finetuning – no matter how small a fraction of some other number it may be” ignores the fact that it is precisely the ratios of these numbers in which finetuning is interested, since it gives an approximation to the degree of finetuning. The argument is not “undermined” by such trivialities as “using instead of renders a value for … a factor of 6.28318 less”. The estimate of the lifepermitting range could be wrong by a thousand, and wrong by a thousand, and the lifepermitting fraction is still 1 in . This calculation can be scrutinized in other ways (uniform probability distribution? A billion years? etc.) but lack of “exact precision” in the formulae doesn’t much matter. The formula is remarkable, not because it is exact, but because such a specific, unusual coupling between EM and gravity seems a priori unlikely, rare in the set of possible universes.
That’s why the inaccuracies over which Klee obsesses don’t bother those “astrophysicists”. Klee quotes physicist after physicist to the effect that the details don’t much matter, but rather than taking the hint, holds doggedly to his conspiracy theory about mathematical malfeasance.
The Hoyle Resonance in Carbon
Klee almost makes a relevant comment regarding the Hoyle resonance, but is scuttled by another highschool physics blunder:
Livio and colleagues reported that, in the context of their computer model, the difference between the two energy levels in question could be increased by 60 keV without destroying the observed cosmic abundances of carbon12 and oxygen16. Sixty thousand electron volts is joules, the thermodynamic temperature equivalent of which … is 696.268 million degrees kelvin. How can a temperature window that wide within which resonant energies can fall count as a case of ‘fine tuning’ that results in energy levels that are ‘just barely’ resonant?
The window is wide because 696.268 million is a really big number? Really?! Need I remind everyone that all physical units depend on an arbitrary convention? (And it’s just kelvin, not degrees kelvin). It is meaningless to describe a physical range with units as wide or narrow. One thing can’t be wide. Something can only be wide relative to something else. That something else, in the case of finetuning, is the set of possible values for a parameter. So, to what should we compare the width of the resonance?
If we compare the lifepermitting width of the resonance to the energy of the resonance, then we have a 0.7% finetuning (whether you measure in eV, J or K). This, as later calculations have shown, is similar to the degree to which the nucleonnucleon force must be finetuned, based on more sophisticated modelling of the carbon nucleus and stars. Ekstrom et al. (2010) calculate that this corresponds to a change in the finestructure constant of one part in , and Epelbaum et al. (2013) calculate that the relevant scale for change in the mass of the light quarks is 23%. If we take the mass of the top quark as an approximation for the possible range of light quark masses, then this is a one in a million finetuning. If we compare to the Planck mass, then it’s one in . This is related to the famous hierarchy problem of particle physics – the mass scales that appear in the standard model of particle physics are unnaturally small. No physicist thinks that the solution to this problem is to convert all your units into kelvin and then shout “Hurray! Big numbers!”.
Blunders Galore
If you’re going to critique some of the finest physicists of the 20th century, you’d better do your homework. I’ll work in Planck units () unless otherwise noted.

In some cases, Klee simply fails to follow CR79’s algebra. Klee complains that the coincidence is off by 4 orders of magnitude. There is a simple reason for this. The full coincidence, from equation (56) of CR79, is , which holds to within a factor of three. CR79 then substitute to eliminate for aesthetic reasons. Being only interested in approximate relations between the constants, they don’t include the factor of 10. If Klee wants to be pedantic, he can keep the extra numerical factor. The coincidence then reads , which holds to about a factor of three and explains where the extra 4 orders of magnitude went. No fudging required.

Table 1 compares CR79’s formulae for the Planck mass and length to the “actual value”. I don’t know where Klee got the “actual value” from, since CR79 gives the definitions of these parameters. The formula , where is the Bohr radius, results not from a finetuning calculation but the decision from CR79 to label their Fig. 1 using lengths in units of . The formula’s inaccuracy comes from the same source as above – they don’t include the factor of 10 when substituting .

That same factor of 10 again accounts for the inaccuracy of the proton length. CR79 give the formula , which is not an order of magnitude calculation but the exact formula for the proton’s reduced Compton wavelength (remember: Planck units). In the label of CR79’s Figure 1, they substitute and leave off the factor of 10, which would put the calculated value within an order of magnitude of the actual value. Also, the “actual value” is taken without justification to be 1 fermi = m. This is backwards: the calculation of the proton Compton wavelength is being used as the approximation, and the nearest SI units order of magnitude as the “actual value”.

No one, to the best of my knowledge, has attempted to base a finetuning claim on an OoM calculation for the mass and height of a human. Such calculations exist to show that, rather remarkably, one can start with the laws of nature and approximately predict the mass and length scales of large animals. The accuracy of these estimates has no bearing on finetuning whatsoever. Also, more modern calculations are correct within an order of magnitude; see Don Page’s wonderful paper “The Height of a Giraffe”.

Where a structure has a range of sizes and masses, and OoM calculation can be considered successful if it gives a typical scale. Given the level of approximation, it is enough for the calculation to fall close to the range of the actual values. It is thus incorrect to compare the calculated typical stellar mass (size) to the mass (size) of the sun alone. Also, Barrow and Tipler (page 332) calculates the range of stellar sizes.

As others have noted, the “actual value” for galaxies is wrong, both for only considering the Milky Way, and for getting the mass and size of the Milky Way wrong.

You really shouldn’t comment on modern cosmology if, when in need of an estimate for the Hubble mass, you draw it from Davies (1982). An awful lot happened in cosmology in the 20 years from 1982 to Klee’s publication in 2002. Klee critiques Davies’ discussion of the finetuning of the energy density of the universe, failing to note that the OoM calculation is too low because the energy density of the universe was not well known in 1982. (The equation in question is, in fact, exact and not an OoM estimate.) More modern measurements show that the present day energy density is within 1% of critical density. Also, the whole point of this finetuning case is that the energy density in the early universe must be finetuned. Even using Davies’ 1982 numbers, the energy density at 1 second (BBN) must be finetuned to about 1 part in , and to 1 part in back at the Planck time (see Appendix A.1 of my paper, or any cosmology textbook, for the relevant calculations). Klee clearly doesn’t understand the flatness problem.

You’re in no place to accuse others of mathematical incompetence when you’re prepared to give statistics like “the maximum imprecision .. is 384.63676”. When physicists get together in tea rooms to chuckle about firstyear physics student blunders, giving 8 significant figures on an OoM calculation causes much mirth. TROP doesn’t even use significant figures consistently; TROP’s Table 1 seems to follow a blanket 5decimalplaces policy.

Even the calculations of the table doesn’t prove Klee’s point. The mean is accuracy is “19.23328”, which is about orders of magnitude. Which is about an order of magnitude.

On a related note, the term “order of magnitude” shouldn’t be interpreted too literally. When one is concerned with approximate calculations, there is no reason to treat “10” as a magic number, a sharp dividing line between precision and deception. Statements like “42.8% of cases are more than one order of magnitude off from exact precision” are thus inappropriate. (And, the statistic 6 out of 14 has a standard error of about 20% so giving 3 significant figures on the percentage is another firstyear statistics mistake.) The required accuracy is determined by context. For example, the coincidence comes from setting the energy of photons emitted by a star to the energy of molecular bonds. This is required for photosynthesis to allow chemical energy to be harnessed from sunlight. The required accuracy is set by the range of molecular bond energies, and the range of EM wavelengths emitted by a star.

Klee wonders whether, when Davies says that , he meant to define in terms of the electron mass, but notes that “nothing in the text suggests that the reader was supposed to switch”. Actually, Davies in the relevant section of “The Accidental Universe” (page 80) refers back to an earlier calculation in section 3.1, where the relevant equation (3.6) shows that is to be used. The same equation is equation (61) of CR79. The relevant coincidence is:
If you’re keeping score, every line on TROP’s Table 1, the centrepiece of the paper, is wrong in some way. Planck – confuses a definition with an order of magnitude calculation, gives incomplete formula for Planck length. Proton – mass is a parameter not an estimate, incomplete formula for length, wrong “actual value”. Humans – not relevant to finetuning as no related anthropic coincidence. Planet – comparison to average instead of range is misleading. Star – comparison of typical star to sun instead of range is misleading, ignores Barrow and Tipler’s calculation of the range. Galaxy – comparison to Milky Way instead of range misleading, actual value for Milky Way is wrong. Hubble – relies on cosmological data 20 years out of date. So, on top of Klee’s mistaken belief that OoM precision is finetuning, most of his examples of OoM imprecision evaporate on close examination. He aims at the wrong target, and misses. There really isn’t much to say for this article at all.
Conclusion
Sean Carroll summarises my point rather nicely:
[I]f your thesis requires that generations of scientists have completely missed some idea that, when you sit and think about it, is really pretty frikkin’ obvious — maybe you should do a little homework before using it as a jumpingoff point for a rant about the intellectual shortcomings of others.
Epilogue
In the course of writing my review paper on finetuning, I had occasion to read some of the work of philosophers of science on such topics as the foundations of general relativity and the role of symmetries in physics, and since then more widely on the philosophy of spacetime and quantum mechanics. I am mightily impressed by the insight into the relevant physics and the clarity of thought that philosophers such as John Norton, John Earman, Katherine Brading, Harvey Brown, Richard Healey, Christopher Martin, Tim Maudlin, David Wallace, David Albert and others bring to their work. The point of this post is not to ridicule philosophy/philosophers of science in general. Rather, consider this a warning of the dangers of failing to heed the wise words of Earman: “… it needs repeating that philosophy of science quickly becomes sterile when it loses contact with what is going on in science”. In Klee’s case, it’s embarrassingly clear that contact was never established in the first place.
The problem lies on both sides. Physicists are among the most prone to blunder into other disciplines, appear foolish and not even know it. Certain physicists and cosmologists have said some remarkably stupid things about philosophy and philosophers recently. We’re not all like that.
Still, I can’t help but wonder how this article passed a referee. Surely, any physicist would immediately spot the ridiculous number of significant figures in Table 1 – we have to correct that kind of mistake in first year labs and tutorials every week. I was under the impression that Brit. J. Phil. Sci. was reasonably prestigious. Is there a shortage of scientists willing to referee philosophy papers? Do any scientists referee philosophy papers? Perhaps such journals should spread their referee net a little wider.
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Footnotes
1. I’ll leave a question mark over Tipler. His technical work is wellrespected but his other views are, at best, highly unorthodox.
EDIT: An earlier version of this post referred to the author as “Richard Klee”, rather than “Robert Klee”. Apologies.
[…] Not So Sharp: A FineTuned Critique of Richard Klee Beginning with Hugh Ross, I undertook to critique various articles on the finetuning of the universe for intelligent life that I deemed to be woeful, or at least in need of correction. A list of previous critiques can be found here. I generally looked for published work, as correcting every blog po … Thu, 20 Jun 2013 21:02:00 CDT more info… […]
Great post! Really nailed it. Klee got me facepalming numerous times.
Btw, the Carter coincidence α(G) ~ α¹²β⁴ can also, interestingly, be derived from the requirement that the habitable zone (using the ‘biological’ temperature of Barrow and Tipler) is not positioned within the Roche limit of the star.
What do you think of Yasunori Nomura’s model?
Hi Luke, I was wondering if I could have your thoughts on this:
http://www.technologyreview.com/view/422444/evidenceemergesthatlawsofphysicsarenotfinetunedforlife/
Olof: interesting. Is that written down anywhere? It seems like this would only be a one sided limit, however: life requires r_habitable > r_roche.
Tayyib: If you follow the link in the article to Page’s paper, you’ll see that he renamed it “Preliminary Inconclusive Hint of Evidence Against Optimal Fine Tuning of the Cosmological Constant for Maximizing the Fraction of Baryons Becoming Life”. There is a difference between finetuning and optimality. Page addresses the second claim. Interesting, nevertheless.
Thanks Luke
[…] of different physical parameters. There are approximations in these calculations – they are orderofmagnitude – but this usually involves assuming that a dimensionless mathematical constant is […]
[…] now applied to different scenarios. I think Maudlin has underestimated both the power of order of magnitude calculations in physics, and the effort that theoretical physicists have put into finetuning calculations. For example, […]