In what follows, I’ll consider Carrier’s claims about the mathematical foundations of probability theory. What Carrier says about probability is at odds with every probability textbook (or lecture notes) I can find. He rejects the foundations of probability laid by frequentists (e.g. Kolmogorov’s axioms) and Bayesians (e.g. Cox’s theorem). He is neither, because we’re all wrong – only Carrier knows how to do probability correctly. That’s why he has consistently refused my repeated requests to provide scholarly references – they do not exist. As such, Carrier cannot borrow the results and standing of modern probability theory. Until he has completed his revolution and published a rigorous mathematical account of *Carrierian probability theory*, all of his claims about probability are meaningless.

I intend to *demonstrate* these claims, so we’ll start by quoting Carrier at length. I won’t be relying on previous posts. In TEC, Carrier says:

Bayes’ theorem is an argument in formal logic that derives the probability that a claim is true from certain other probabilities about that theory and the evidence. It’s been formally proven, so no one who accepts its premises can rationally deny its conclusion. It has four premises … [namely P(h|b), P(~h|b), P(e|h.b), P(e|~h.b)]. … Once we have [those], the conclusion necessarily follows according to a fixed formula. That conclusion is then by definition the probability that our claim h is true given all our evidence e and our background knowledge b.

In OBR, he says:

[E]ver since the Principia Mathematica it has been an established fact that nearly all mathematics reduces to formal logic … The relevant probability theory can be deduced from Willard Arithmetic … anyone familiar with both Bayes’ Theorem (hereafter BT) and conditional logic (i.e. syllogisms constructed of if/then propositions) can see from what I show there [in Proving History] that BT indeed is reducible to a syllogism in conditional logic, where the statements of each probability-variable within the formula is a premise in formal logic, and the conclusion of the equation becomes the conclusion of the syllogism. In the simplest terms, “if P(h|b) is w and P(e|h.b) is x and P(e|~h.b) is y, then P(h|e.b) is z,” which is a logically necessary truth, becomes the concluding major premise, and “P(h|b) is w and P(e|h.b) is x and P(e|~h.b) is y” are the minor premises. And one can prove the major premise true by building syllogisms all the way down to the formal proof of BT, again by symbolic logic (which one can again replace with old-fashioned propositional logic if one were so inclined).

More specifically it is a form of argument, that is, a logical formula that describes a particular kind of argument. The form of this argument is logically valid. That is, its conclusion is necessarily true when its premises are true. Which means, if the three variables in BT are true (each representing a proposition about a probability, hence a premise in an argument), the epistemic probability that results is then a logically necessary truth. So, yes, Bayes’ Theorem is an argument.

He links to, and later shows, the following “Proof of Bayes Theorem … by symbolic logic”, saying that “the derivation of the theorem is *this*.”

For future reference, we’ll call this **“The Proof”. **Of his mathematical notation, Carrier says:

P(h|b) is symbolic notation for the proposition “the probability that a designated hypothesis is true given all available background knowledge but not the evidence to be examined is x,” where x is an assigned probability in the argument.

I have 13 probability textbooks/lecture notes open in front of me: Bain and Engelhardt; Jaynes (PDF); Wall and Jenkins; MacKay (PDF); Grinstead and Snell; Ash; Bertsekas and Tsitsiklis; Rosenthal; Bayer; Dembo; Sokol and Rønn-Nielsen; Venkatesh; Durrett; Tao. I recently stopped by Sydney University’s Library to pick up a book on nuclear reactions, and took the time to open another 15 textbooks. I’ve even checked some of the philosophy of probability literature, such as Antony Eagle’s collection of readings (highly recommended), Arnborg and Sjodin, Caticha, Colyvan, Hajek (who has a number of great papers on probability), and Maudlin.

When presenting the foundations of probability theory, these textbooks and articles roughly divide along Bayesian vs frequentist lines. The purely mathematical approach, typical of frequentist textbooks, begins by thinking about relative frequencies before introducing measure theory, explaining Kolmogorov’s axioms, motivating the definition of conditional probability, and then – in one line of algebra – giving “The Proof” of Bayes theorem. Says Mosteller, Rourke and Thomas: “At the mathematical level, there is hardly any disagreement about the foundations of probability … The foundation in set theory was laid in 1933 by the great Russian probabilitist, A. Kolmogorov.” With this mathematical apparatus in hand, we use it to analyse relative frequencies of data.

Bayesians take a different approach (e.g. Probability Theory by Ed Jaynes). We start by thinking about modelling degrees of plausibility. The frequentist, quite rightly, asks what the foundations of this approach are. In particular, why think that degrees of plausibility should be modelled by probabilities? Why think that “plausibilities” can be mathematised at all, and why use Kolmogorov’s particular mathematical apparatus? Bayesians respond by motivating certain “desiderata of rationality”, and use these to prove via Cox’s theorem (or perhaps via de Finetti’s “Dutch Book” arguments) that degrees of plausibility obey the usual rules of probability. In particular, the produce rule is proven, p(A and B | C) = p(A|B and C) p(B|C), from which Bayes theorem follows via “The Proof”.

*In precisely none of these textbooks and articles will you find anything like Carrier’s account. *When presenting the foundations of probability theory in general and Bayes Theorem in particular, no one presents anything like Carrier’s version of probability theory. Do it yourself, if you have the time and resources. Get a textbook (some of the links above are to online PDFs), find the sections on the foundations of probability and Bayes Theorem, and compare to the quotes from Carrier above. In this company, Carrier’s version of probability theory is a total loner. We’ll see why.

To draw out the various idiosyncrasies of Carrier’s account of Bayes Theorem, consider this parallel discussion of a different mathematical theorem:

*Pythagoras Theorem (PT) is an argument in formal logic. It’s been formally proven. It follows from two premises (a and b), from which the conclusion (c) follows according to a fixed formula, where a and b are assigned a value in the argument. PT is reducible to a syllogism in conditional logic as follows:*

*(1′) If the two shorter sides of a right-angled triangle are a and b, then the hypotenuse is c.*

*(2′) The shorter sides of a right-angled triangle are a and b*

*(3′) Therefore, the hypotenuse is c*

*One can prove (1′) by building syllogisms all the way down to the formal proof of PT. So, Pythagoras theorem is an argument, or a form of argument. Its conclusion (c) is necessarily true when its premises (a and b) are true.*

The problems are legion.

- This “argument form” of PT is missing PT itself; we must add “where c
^{2}= a^{2}+ b^{2}” to premise (1′) to give it any meaning. - It does not show that Pythagoras theorem
**is**an argument or a form of argument. It shows that PT can be used**in**an argument. But that’s trivial – any statement can be a premise in an argument. - The “form” of the argument is just
*modus ponens*, “If A then B. A. Therefore B”. There is no particular “form of argument” associated with PT. - Don’t call a, b and c “premises” or “symbolic notation for a proposition”. You can’t multiply and add premises or propositions, and that’s what happens in PT. They’re numbers. You put them in a formula.
- The discussion above is not a proof of PT because PT is (or should be; see A) included in premise (1′). Nor does it show how PT follows from an axiomatization of mathematics or reduces to symbolic logic. It shows how to argue
**from**PT, not**to**PT. - You cannot prove PT from the axioms of arithmetic because those axioms don’t define what a triangle and a right angle are, or what to do with them. You need axioms of plane geometry, such as Euclid’s axioms (or their more modern descendants).

We can apply A-F straightforwardly to Carrier’s discussion of BT. For convenience, I’ll number Carrier’s premises:

(1) If P(h|b) is w and P(e|h.b) is x and P(e|~h.b) is y, then P(h|e.b) is z

(2) P(h|b) is w and P(e|h.b) is x and P(e|~h.b) is y

(3) Therefore, P(h|e.b) is z

- This “argument form” of BT is missing BT itself. As it stands, (1) states that P(h|e.b) is equal to some unspecified, arbitrary number. We must add “where z = xw / [xw + y(1 – w)]” to premise (1) to give it any meaning.
- It does not show that Bayes theorem
**is**an argument or a form of argument. It shows that BT can be**used**in a syllogism … as can any other statement. It is also an unnecessary complication to use BT in this form – what Carrier calls the “reduction to a syllogism in conditional logic” every mathematician would call “putting numbers into a formula”. - The “form” of the argument is just
*modus ponens*, “If A then B. A. Therefore B”. There is no particular “form of argument” associated with BT. - Don’t call P(e|h.b) etc premises or “symbolic notation for [a] proposition”, because you can’t multiply and add and divide premises and propositions, and that’s what happens in BT. They’re numbers. You put them in a formula.
- The syllogism (1)-(3) is not a proof of BT, because BT is (or should be; see A) included in premise (1). Nor does it show how BT follows from an axiomatization of mathematics or reduces to symbolic logic. It shows how to argue
**from**BT, not**to**BT. - You cannot prove BT from the axioms of arithmetic because they don’t know what a probability is, or what to do with it. Read, for example, the Peano axioms. They define natural numbers, equality, succession – but not probability. You need the axioms of probability theory, such as Kolmogorov’s axioms.

All of which brings us to “The Proof”, which is nothing of the sort. It is an elementary probability exercise, the kind of thing you’d set a first year student: show that Bayes theorem follows from the product rule (Statement (1) in “The Proof”). Actually, the problem is so easy so most textbooks just do it in one line and move on. Venkatesh (page 56), for example, presents “The Proof” in one line and says that it is “… little more than a restatement of the definition of conditional probability.”

Showing that a statement follows from some other statement does not prove it. You have to show that it follows from the relevant axioms, or a theorem derived from those axioms. ** Given** that the product rule follows from the axioms of probability theory or Cox’s theorem or from the definition of conditional probabilities (which it does), “The Proof” does in fact establish BT. But Carrier is claiming more: “The Proof” is supposedly “the formal proof of BT … by symbolic logic”, showing how this mathematical theorem “reduces to formal logic” in the rigorous tradition of the Principia Mathematica.

This is just wrong. “The Proof” is a derivation of BT from the product rule. We’re a million miles from the axiomatic foundations of mathematics. Worse, Carrier thinks that “The Proof” is “by symbolic logic”, when it is quite plainly an exercise in algebra. The product rule involves the logical operator “and”, but all the manipulations are algebraic (multiplying and dividing), not logical. I’ll repeat that point: *this supposed proof “by symbolic logic” uses none of the rules of symbolic logic*.

This is not a small technicality. A trivial algebraic exercise, too easy for any competent student, is being presented by Carrier as a rigorous, formal, first-principles proof of Bayes theorem using symbolic logic alone. This is decisive evidence of Carrier’s utter cluelessness when it comes to probability theory. No mathematician, when asked about the foundations of probability theory, will point to “The Proof”.

Point F is perhaps the most important, so I’ll expand on it. As we saw above, there is a substantial academic literature on the foundations of probability, the status of Bayes Theorem, its relation to the interpretation of probability, and the various ways in which it can be derived. Generally, Bayesians go with Cox’s theorem or Dutch book arguments, while frequentists go with Kolmogorov.

** Carrier, all out on his own, needs none of this**. He is not using the established results of modern probability theory. He isn’t really a Bayesian or a frequentist or anything that mathematicians have seen before. He has his own,

In fact, in *Proving History* he makes these claims explicit. In the section “Bayesianism as Epistemic Frequentism” (Chapter 6), he outlines an approach to probabilities, according to which “** all Bayesians are in fact frequentists**“. When Bayesians claim that probabilities are not frequencies “

What is this frequency that all Bayesians everywhere have been ignoring all this time? According to Carrier, degrees of belief are really the ratio of the number of “beliefs that are true” to the number of “all beliefs backed by a certain comparable quantity and quality of evidence”. For example, “When a Bayesian says that the prior probability that a royal flush is fair is 95% … they are *really* saying that 95% of all royal flushes drawn (on relevantly similar occasions) are fair. Which is a physical frequency. Thus, epistemic probabilities always derive from physical probabilities.”

It may seem ungrateful to nit-pick such a monumental intellectual achievement, but he’s not quite done. To complete his exposition of Carrierian probability, and show those so-called mathematicians how probability should be done, Carrier has one more task: justify a ** mathematical method** that allows us to

Heck, I’ll even get him started. Perhaps, to allow for comparison, “quality and quantity” could be represented by a real numbers q. More evidence could mean larger q values, and adding evidence should never decrease q. We should stipulate consistency: equivalent states of evidence receive equal q values. In which case, good news! Just such a method already exists! The principles that govern q are basically the premises of Cox’s theorem, from which we get Bayesian probabilities.

So let’s review Carrier’s method. To calculate probabilities, we need to define the reference classes in which to place our various beliefs. And to define those, we’ll need a * measure* of the quantity and quality of evidence. This measure, it turns out, is basically Bayesian probabilities.

Carrierian probability, as presented, is thus hopelessly incomplete. And when completed, circular. The circularity is staring us in the face when Carrier says that the relevant reference class, from which we calculate probabilities, is defined as containing those beliefs “backed by the kind of evidence and data that produces those kinds of prior and consequent probabilities.” Only once we have probabilities can we form such reference classes (ignoring the issue of when two “amounts of evidence” are “comparable” – exactly equal? Within 1%?). It follows that these reference classes cannot be used to define the probabilities.

So, it’s time to put up or shut up. When invited to outline the foundations of his approach to probability theory (which I first did 2 years ago – question 5), Carrier snubs modern probability theory. Axioms, Kolmogorov, Cox, de Finetti … who needs them! But a mathematical theory without foundations is just hot air. Until a rigorous basis for Carrierian probability theory is provided, all his probability claims are meaningless.

Carrier must publish a **series of papers** in mathematical journals that substantiate his extraordinary claims about the foundations of probability theory, proving – in the face of centuries of work by mathematicians – that:

- None of the usual axioms or arguments or theorems are needed.
- Probability reduces to formal/symbolic logic
*alone.* - Bayesianism is really a kind of frequentism.
- “Quality and quantity of evidence” can be uniquely and precisely quantified.

He made these claims, so this is a task that he has set for himself. Citing his own books is not good enough.

Or else, we’ll know that he is all bluff and bluster. Pressed to present the foundations of probability theory, he has failed utterly. He could have just plagiarised any probability textbook, but instead invented a pile of garbage about conditional logic, building syllogisms, *Principia Mathematica*, the axioms of arithmetic, and quality of evidence.

Hence, this is my final word. If Carrierian probability is hailed as a revolution by mathematicians, then I will concede Carrier’s probabilistic credentials and be forever silenced. If he continues to talk about probabilities, then – since he doesn’t mean by “probability” what the term means in any rigorous mathematical theory – this must be regarded as literally meaningless. We need only reply: **where are the papers?**

If, alternatively, he realises that he is completely out of his depth, that he hasn’t got the first clue about the foundations of probability theory, he may (after learning probability theory – for the first time, it seems – from a textbook) try to claim that he has been a follower of Cox/Kolmogorov all along. However, as we have seen, this is complete shift in the foundations of his approach. All of his previous work that relies on Carrierian probability – including its extension to historical investigation in *Proving History* and *On the Historicity of Jesus* – must be discarded.

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Note that this is actually not “my” conclusion. It is the conclusion of three mathematicians (including one astrophysicist) in two different studies converging on the same result independently of each other.

Wow! Two “studies”! (In academia, we call them “papers”. Though, neither were published in a peer-reviewed journal, so perhaps “articles”.) Three mathematicians! Except that Elliott Sober is a philosopher (and a fine one), not a mathematician – he has never published a paper in a mathematics journal. More grasping at straws.

Barnes wants to get a different result by insisting the prior probability of observers is low—which means, because prior probabilities are always relative probabilities, that that probability is low without God, i.e. that it is on prior considerations far more likely that observers would exist if God exists than if He doesn’t.

Those sentences fail Bayesian Probability 101. Prior probabilities are probabilities * of hypotheses*. Always. In every probability textbook there has ever been

This is not a harmless slip in terminology. Carrier treats a likelihood as if it were a prior. He has confused the ** concepts**, not just the names. Carrier states that “the only way the prior probability of observers can be low, is if the prior probability of observers is high on some alternative hypothesis.”

It follows that this entire section on the “prior probability of observers” and the need to consider “some alternative hypothesis” is garbage. There is simply no argument to respond to, only a hopeless mess of Carrier’s confusions. It’s an extended discussion about prior probabilities from a guy who doesn’t know what a prior probability is. Given that he has previously confused priors and posteriors, he’s zero from three on the fundamentals of Bayes theorem. You cannot keep getting the basics of probability theory wrong and expect to be taken seriously.

**Technical details: **For any hypothesis h, and its negation ~h (which we can think of as the disjunction or union of all alternatives to h), p(h|b) +p(~h|b) = 1. So, the prior p(h|b) is small if and only if p(~h|b) is large, and vice versa. The same applies to posteriors: p(h|e.b) +p(~h|e.b) = 1. But there is no corresponding rule for likelihoods and hypotheses: p(e|h.b) is small does not imply that p(e|~h.b) is large. “p(e|h.b) + p(e|~h.b) = 1” is * not* an identity of probability theory.

This is where note 23 in my chapter comes in … Barnes never mentions this argument and never responds to this argument.

Addressed in Part 2, under “Bayes’ Theorem Omits Redundancies” and Part 4, under “The Main Attraction” and “My Reply”. I’ve put Carrier’s argument in mathematical notation, so it should be easy to demonstrate where my response falls short. No such demonstration is forthcoming, only repetition.

… [when you] remove even our knowledge of ourselves existing from b [the background evidence]. You end up making statements about universes without observers in them. Which can never be observed. … Either you are making statements about universes that have a ZERO% chance of being observed (and therefore cannot be true of our universe), or you are making statements that are 100% guaranteed to be observed.

This is exactly the point I discussed in detail in Part 4. Since Bayes theorem is an identity – that is, it can be used with *any* propositions – moving a particular fact between e and b can never be wrong. Carrier’s objections must be mistaken, since you can’t fight a mathematical identity.

And we can see where they are mistaken. In Bayesian probability theory, hypotheses are penalised for declaring as “highly likely” statements that are in fact false. For example, the hypothesis “the burglar guessed the 12-digit combination to the safe” implies that it is highly likely that the burglar didn’t open the safe. It is heavily penalised, then, if security camera footage shows the burglar opening the safe on the first attempt. We end up talking about burglars who didn’t open the safe because those kind of burglars are the most likely on the stated hypothesis.

If naturalism implies that, given * only* that a universe exists, it is highly likely that the universe does not contain life forms, then it is heavily penalised by the falsity of that statement. (We all understand background information, right?) We end up talking about universes without observers because those kind of universes are the most likely on naturalism. The fact that they cannot be observed does not matter; likelihoods are normalised over an

Let’s recap some highlights of these three posts.

- Carrier has not addressed the charge of inconsistency with probability theory. In fact, he given more examples of inconsistency by introducing “hypothetical reference classes”. He has not addressed the reference class problem.
- He has made up probability concepts that no one has ever heard of before, including “transfinite frequentism” and “existential probability calculus”.
- He has abandoned his previous claim that “all the scientific models we have … show life-bearing universes to be a common result of random universe variation, not a rare one.”
- He completely misunderstands my rather obvious point that “for a given possible universe, we specify the physics”, and in so doing, shows that he does not understand fine-tuning at its most basic level.
- And, finally, Carrier’s argument regarding the “Real Heart of the Matter” is rendered meaningless by a deep misunderstanding of probability theory’s basics.

Carrier, demonstrably, understands neither probability theory nor fine-tuning.

Barring a minor miracle, my next post will be my last about Richard Carrier. I’ll explain why there.

- I’m taking the term “hypotheses” in a general sense, so that it could include the hypothesis that an unknown parameter has a certain value. That is, priors can be distributions of unknown parameters.
- This talk of “some alternative hypothesis” precludes the possibility that Carrier is actually referring to p(e|b), the marginal likelihood. If “e = this universe contains observers”, then p(e|b) could – I suppose – be referred to as the prior probability of observers, though no one would and Carrier’s argument would still be wrong.

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**Location:** Western Sydney University, Lecture theatre, Building 30

**Date:** Monday 15th February, 7.30 pm

**Title:** There is more to the Universe than its good looks.

**Abstract: **The planets, stars and galaxies that fill the night sky obey elegant mathematical patterns: the laws of nature. Why does our Universe obey these particular laws? As a clue to answering this question, scientists have asked a similar question: what if the laws were slightly different? What if it had begun with more matter, had heavier particles, or space had four dimensions?

In the last 30 years, scientists have discovered something astounding: the vast majority of these changes are disastrous. We end up with a universe containing no galaxies, no stars, no planets, no atoms, no molecules, and most importantly, no intelligent life-forms wondering what went wrong. This is called the fine-tuning of the universe for life. After explaining the science of what happens when you change the way our universe works, we will ask: what does all this mean?

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Barnes claims to have hundreds of science papers that refute what I say about the possibility space of universe construction, and Lowder thinks this is devastating, but Barnes does not cite a single paper that answers my point.

My comment was in response to the claim that the statement “the fundamental constants and quantities of nature must fall into an incomprehensibly narrow life-permitting range” has been “refuted by scientists”, not about what Carrier has to say about “universe construction”. The references are in my review paper.

Because we don’t know how many variables there are.

Carrier doesn’t – he still thinks that there are 6 fundamental constants of nature, but can’t say what they are. Actual physicists have no problem counting the free parameters of fundamental physics as we know it, which is what fine-tuning is all about.

We don’t know all the outcomes of varying them against each other.

We know enough, thanks to a few decades of scientific research. It is not an argument from ignorance – extensive calculations have been performed, which overwhelmingly support fine-tuning.

And, ironically for Barnes, we don’t have the transfinite mathematics to solve the problem.

This is probably a reference to “transfinite frequentism”, a term that, as we saw last time, Carrier invented.

In any case, we don’t need transfinite arithmetic here. Bayesian probability deals with free parameters with infinite ranges in physics all the time; fine-tuning is not a unique case. Many of the technical probability objections aimed at fine-tuning, such as those of the McGrews, would preclude a very wide range of applications of probability in physics.

I am not aware of any paper in cosmology that addresses these issues.

It’s called the “measure problem”. There are literally hundreds of papers on it, too. For example, here’s a relevant paper with over 100 citations: “Measure problem in cosmology“. Aguirre (2005), Tegmark (2005), Vilenkin (2006) and Olum (2012) are good places to start. The problem of infinities in cosmology (including in fine-tuning and the multiverse) is tricky, but few cosmologists believe that it is unsolvable.

In this case, it’s not even an argument in my chapter in TEC … Barnes has skipped to quoting and arguing against a completely unrelated blog post of mine.

My third post discusses a post of Carrier’s that a) discusses fine-tuning and b) quotes from TEC. Unrelated? Grasping at straws …

“We actually do not know that there is only a narrow life-permitting range of possible configurations of the universe.” Barnes can cite no paper refuting that statement.

We “do not know” only in the trivial sense that we aren’t *completely, 100% certain*, but almost the entire fine-tuning literature is evidence against that statement. For example, read just about any paper on the cosmological constant problem: Hartle et al. (2013) “Anthropic reasoning potentially explains why the observed value of the cosmological constant is small when compared to natural values set by the Planck scale as was discussed by Barrow and Tipler, and Weinberg.”

… some studies get a wide range not a narrow one … e.g. Fred Adams, “Stars in Other Universes: Stellar Structure with Different Fundamental Constants”

Adams does not get a wide range – the figure of “one fourth” mentioned in the abstract is not a measure of the life-permitting range. See this post, and my comments in the review paper.

… which suggests to me he is not being honest in what he claims to know about the literature. So we have inconsistent results.

I’m dishonest because I didn’t mention the “handful [of papers] that oppose this conclusion [of fine-tuning]”? Wait … that’s a quote from me. The results are anything but “inconsistent”.

Speaking of inconsistency, In his post from 2013, Carrier says “all the scientific models we have … show life-bearing universes to be a common result of random universe variation, not a rare one.” Now, in OBR, he says “some studies get a wide range not a narrow one … I know they exist, because I’ve read more than one.”

Then I go on to give the second reason, which is that even those papers are useless. Notice Barnes does not tell his readers this. … my very next sentence, the sentence Barnes hides until later. [Barnes] prefers to pretend [the second argument] didn’t exist than attempt to answer it.

Note the inconsistency between “doesn’t tell” and “hides until later”. Also, note my diabolical method of “hiding” Carrier’s arguments by quoting them. Read my post: I discuss the first reason. And then I *immediately* discuss the second reason, saying “For a given possible universe, we specify the physics. So we know that there are no other constants and variables.”

Carrier later responds to my reply. So he wants to complain that a) I didn’t respond and (b) my response is mistaken. You can’t have it both ways.

As an aside, if we want to talk about dishonesty: In his post and OBR (linked from “or the mathematical problem”), Carrier cites Tim and Lydia McGrew in support of his claim that infinities create serious problems for fine-tuning. What he doesn’t tell you is that Lydia describes Carrier as “styling himself some sort of probability expert” and “show[ing] a rather striking lack of understanding of probability”. Tim, meanwhile, has shown that Carrier’s attempts to teach basic probability theory are riddled with elementary errors, demonstrating that “Richard Carrier is completely out of his depth with respect to the mathematics of elementary probability. He garbles the explanation of elementary concepts, and he fumbles the computation of his own chosen examples. … Carrier has not crossed the *pons asinorum* of elementary probability. … Why on earth would anyone take Richard Carrier seriously on this topic when he’s shown himself to be wildly incompetent?”. I couldn’t agree more. Tim’s “Does Richard Carrier Exist?” is also well worth a read.

Does Carrier tell his readers this? Hostile witnesses, who admit something against their own biases, are fine, of course. But is it honest (or, indeed, a good idea) to cite, in support of your case, experts who think that you are wildly incompetent and cannot be taken seriously?

Lowder appears to have been duped by Barnes into thinking I said it was a fact now that “the number of configurations that are life permitting actually ends up respectably high (between 1 in 8 and 1 in 4…).” Nope. Because my very next sentence, the sentence Barnes hides until later, and pretends isn’t a continuation of the same argument, says: “And even those models are artificially limiting the constants that vary to the constants in our universe, when in fact there can be any number of other constants and variables.

Sorry, Jeff – my Jedi mind tricks must be better than I realised. What Carrier claimed was “When you allow all the constants to vary freely, the number of configurations that are life permitting actually ends up respectably high (between 1 in 8 and 1 in 4).” That claim is false. That Carrier has *more to say* does not excuse his mistake.

Carrier presents my response to his second argument as:

Walk through the thinking here. We know there cannot (!) be or ever have been or ever will be a different universe with different forces, dimensions, and particles than our universe has, because “we specify the physics” (Uh, no, sorry, nature specifies the physics; we just try to guess at what nature does and/or can do) and because “A universe with other constants would be a different universe.” WTF? Um, that’s what we are talking about … different universes! I literally cannot make any sense of Barnes’s argument here.

Yeah, no kidding. What I said was “For a given possible universe, we specify the physics”. It is manifestly not a claim about what cannot ever have been, or about what nature actually does, or about ** actual** universes other than ours, or that our universe could not have or does not have physics of which we are currently unaware. This is not a discussion of the multiverse. The context is the claim that “there is only a narrow life-permitting range of possible configurations of the universe”. A universe with different constants would be a different “possible configuration”.

Moreover, my claim is obviously about other possible universes. All fine-tuning claims are. Carrier’s huff and puff about “if Barnes has some fabulous logical proof that universes with different forces, dimensions, and particles than ours are logically impossible … ” is not just ridiculous. I said “In all the possible universes we have explored, we have found that a tiny fraction would permit the existence of intelligent life.” To misunderstand this point is to completely misunderstand not only what I wrote, but the most basic, definitional claims of fine-tuning.

So there’s an useful conclusion: *Carrier has not critiqued fine-tuning, because he does not know what it is.*

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(I don’t mind the delay. We’re all busy. I’ve still got posts I began in 2014 that I haven’t finished.)

First, a few short replies. I’ll skim through Carrier’s comments and provide a few one(-ish)-line responses. I’m assuming you’ve read Carriers’s post, so the quotes below (from OBR unless otherwise noted) are meant to point to (rather than reproduce) the relevant section. My discussion here is incomplete; later posts will go into more detail.

Carrier notes that his argument is a popularisation of other works, saying later that “Barnes … ignores the original papers I’m summarizing.”

I’ve responded to Ikeda and Jeffrey’s article here and here. Their reasoning is valid, but is not about fine-tuning. I show how the fine-tuning argument, properly formulated, avoids their critique. My response to Sober would be similar.

Lowder agrees with Barnes on a few things, but only by trusting that Barnes actually correctly described my argument. He didn’t.

The first of umpteen “Barnes just does understand me” complaints. The reader will have to decide for themselves. Note both the numerous lengthy quotes I typed out in my posts, and my many attempts to formulate Carrier’s arguments in precise, mathematical notation.

On the general problem of deriving frequencies from reference classes, Bayesians have written extensively.

Deriving frequencies from reference classes is trivial – you just count members and divide. The problem that references classes create for finite frequentism is their definition, not how one counts their members. So, Carrier doesn’t understand the reference class problem.

This last is the more bizarre gaffe of his, because calculating the range of possible universes is a routine practice in cosmological science.

What I said was “The restriction to known, actual events creates an obvious problem for the study of unique events.” Bayesians can apply probability to the universe; finite frequentists can’t. That’s why most cosmologists are Bayesians.

Our universe is not the only logically possible one to have arisen. That in fact it is not sitting in a reference class of one, but a reference class of an infinite number of configurations of laws and constants.

Keep clearly in mind my claim in Part 1: Carrier’s approach to probability is inconsistent. He keeps shifting the goalposts. In TEC, when talking about a cosmic designer, he says “Probability measures frequency (whether of things happening or of things being true)”. Only known cases, verified by science, can be allowed in a reference class. But now, in OBR, it’s OK to put hypothetical possibilities in a reference class.

This destroys his argument on page 282-3 of TEC, in which Carrier distinguishes cases that science has verified from “alleged cases”, which must be excluded from the reference class. But alleged cases are logically possible, so they should have been included all along, according to OBR.

Barnes would notice that if he didn’t also repeatedly confuse my estimating of the prior (at 25% “God created the universe”) with the threshold probability of coincidences (a distinction I illustrated with the “miraculous machinegun” argument I discuss, a discussion Barnes never actually interacted with, in TEC, pp. 296-98).

My discussion is in Part 2, “The Firing Squad Machine” and following. I quote from Carrier’s essay at length, put Carrier’s argument into standard probability notation, and show that it is invalid. By confusing priors and posteriors, Carrier is not updating in the Bayesian way.

Barnes attacked what I addressed in the chapter as the “threshold” probability discussed in note 31 … [a complete reproduction of the note 31] … This argument Barnes never rebutted.

I never attacked that argument, because the Bayesian approach doesn’t need a probability “threshold”. Dembski’s approach is pure frequentism. His threshold applies to likelihoods; Dembski, as a frequentist, doesn’t believe in priors. As a Bayesian, I agree with Carrier that this approach is flawed. The footnote is not rebutted because there is nothing for the Bayesian to rebut.

In short, since the only universes that can ever be observed (if there is no God) are universes capable of producing life, if only fine tuned universes are capable of producing life, then if God does not exist, only fine tuned universes can ever be observed. This counter-intuitively entails that fine-tuning is 100% expected on atheism.

Again … A fine-tuned universe is 100% expected on atheism if and only if observers are 100% expected on atheism. Observers are not 100% expected on atheism, because most possible universes do not support observers – that’s the point of fine-tuning. Thus, a fine-tuned universe is not 100% expected on atheism. I formalise this argument in Part 4.

Not only did I never argue my chapter’s conclusion from a multiverse, I explicitly said I was rejecting the existence of a multiverse for the sake of a fortiori argument. That Barnes ignored me, even though I kept telling him this, and he instead kept trying to attack some argument from multiverses.

As Lowder’s quotes [6] and [7] demonstrate, I never contend that Carrier argues the “chapter’s conclusion from a multiverse”. Rather, Carrier’s discussion of the multiverse uses a different approach to probability, one that is inconsistent with the approach to probability applied to fine-tuning elsewhere in TEC. This inconsistency undermines his entire approach – the goalposts shift at will.

Because this is where Barnes flips his lid about “finite” frequentism (in case you were wondering what that was in reference to). Note I at no point rely on transfinite frequentism in the argument of my chapter

There is no such thing as “transfinite frequentism”. Take a moment to Google that phrase – the only result is Carrier’s blog post (and possibly now this one). Literally no one ever – no mathematician,no scientist, no philosopher … not even a clueless quack – has ever used that phrase before, so far as Google (and Google Scholar, Google Books, Wikipedia, arxiv.org, Bing, Yahoo!, and even Ask Jeeves) can tell. Draw your own conclusion.

The two kinds of frequentism are called “finite frequentism” and “hypothetical frequentism”. See, for example, the entry “Interpretations of Probability” at SEP, and these two MUST READ critiques by Alan Hajek: “Fifteen Arguments against Finite Frequentism” and “Fifteen Arguments Against Hypothetical Frequentism“.

This statement [“If we are using Bayes’ theorem, the likelihood of each hypothesis is extremely relevant”] simply repeats what I myself argue in my chapter in TEC. Illustrating how much Barnes is simply not even interacting with that chapter’s actual argument.

Again, my problem is that Carrier’s approach is inconsistent. He *says* that likelihoods are relevant, but abandons this principle when convenient. See Part 1 under “Forgetting Bayes’ Theorem” to see an **example** of this inconsistency. Restating the principle does not answer my charge.

Therefore life will never observe itself being in any other kind of universe than one that’s fine tuned. … Barnes to this day has never responded to it.

As above, and in Part 4, under “The Main Attraction”. If my mathematical formulation is in error, then correct it.

… the example proves to us that fine tuning never entails [intelligent design]. To the contrary, every randomly generated universe that has life in it will be finely tuned. That is what the example illustrates. Therefore, in cosmology, there is no meaningful correlation between fine tuning and intelligent design.

No entailment, therefore no correlation. In the context of a probabilistic argument.

Part 2 is here. Part 3 is here.

Filed under: Uncategorized ]]>

In 1971, Freeman Dyson discussed a seemingly fortunate fact about nuclear physics in our universe. Because two protons won’t stick to each other, when they collide inside stars, nothing much happens. Very rarely, however, in the course of the collision the weak nuclear force will turn a proton into a neutron, and the resulting deuterium nucleus (proton + neutron) is stable. These the star can combine into helium, releasing energy.

If a super-villain boasted of a device that could bind the *diproton* (proton + proton) in the Sun, then we’d better listen. The Sun, subject to such a change in nuclear physics, would burn through the entirety of its fuel in about a second. Ouch.

A very small change in the strength of the strong force or the masses of the fundamental particles would bind the diproton. This looks like an outstanding case of find-tuning for life: a very small change in the fundamental constants of nature would produce a decidedly life-destroying outcome.

However, *this is not the right conclusion*. The question of fine-tuning is this: how would the universe have been different if the constants of nature had different values? In the example above, we took *our universe* and abruptly changed the constants half-way through its life. The Sun would explode, but would a bound-diproton universe create stars that explode?

In my review paper, I reported a few reasons to suspect that the diproton disaster isn’t as clear cut as we think, but noted that detailed calculations have not been performed. I’m a cosmologist/galaxy formation kind of astrophysicist, and so hoped that someone else would do it! However, a talk by Mark Krumholz (UC Santa Cruz) showed the way forward. Stars in our universe have an initial deuterium-burning phase, where they burn leftover deuterium from the big bang. They only have a very small amount, but this reaction is very similar to diproton burning in alternative universes.

So I investigated stars that are initially 50% protons, 50% deuterium, and so are primed to burn via the strong force. The result: as expected, stars don’t explode. They simply burn at a lower temperature, and with less dense cores. In particular, for stars with the same total mass, there is only a factor of three difference in the total energy output per unit time. This means that their lifetimes are also similar.

Looking over all the stars available in parameter space – weak burning and strong burning – the most interesting constraint for a life-permitting universe is the maximum stellar lifetime. The figure below shows the strength of electromagnetism (horizontal axis) and the strength of gravity (vertical axis).

Below the dashed lines, hydrogen-burning stars are stable. Below the thick black line, deuterium/diproton burning stars are stable – this is a much larger region. Our universe is the black square. *Note the logarithmic scale!* The contour lines show the lifetime of the longest-lived (and hence smallest) stable star in a given universe. The line labelled “6” shows where the longest-lived star burns out in a million years – too short for planets and life and such. Binding the diproton does not affect chemistry, or indeed any of the physics that upon which living things directly rely.

So, if the strength of gravity () were not very small (< 10^30), all stars would burn out too quickly.This is a conservative but very robust anthropic constraint. Actually, the “strength of gravity” is the ratio of the proton to the Planck mass, so the relevant fine-tuning is the fact that the fundamental particles of nature are “absurdly light“, in the words of Leonard Susskind. These are some of the most important fine-tuning examples around.

Filed under: Astronomy, fine tuning, Physics, The Universe ]]>

- Of Nothing
- More Sweet Nothings
- A universe from nothing? What you should know before you hear the Krauss-Craig debate
- Did the Universe Have a Beginning? – Carroll vs Craig Review (Part 1)

Filed under: Uncategorized ]]>

Tyson’s thesis is as follows:

… a careful reading of older texts, particularly those concerned with the universe itself, shows that the authors invoke divinity only when they reach the boundaries of their understanding.

To support this hypothesis, Tyson quotes Newton, 2nd century Alexandrian astronomer Ptolemy and 17th century Dutch astronomer Christiaan Huygens. The remarkable thing about Tyson’s article is that none of the quotes come close to proving his thesis; in fact, they prove the opposite.

Tyson is quotes from Newton’s General Scholium, an essay appended to the end of the second and third editions of the *Principia.*

But in the absence of data, at the border between what he could explain and what he could only honor—the causes he could identify and those he could not—Newton rapturously invokes God:

“Eternal and Infinite, Omnipotent and Omniscient; … he governs all things, and knows all things that are or can be done. … We know him only by his most wise and excellent contrivances of things, and final causes; we admire him for his perfections; but we reverence and adore him on account of his dominion.”

To be blunt, what part of “he governs all things” doesn’t Tyson understand? God’s “dominion” – the extent of his rule – is “always and everywhere”. Clearly, Newton is not invoking God only at the edge of scientific knowledge, but everywhere and in everything. The Scholium is not long, so I invite you to read it; you will nowhere find Newton saying that God is only found where science has run out of answers. You will find him saying (echoing Paul) that “In him are all things contained and moved.”

When Newton states that “We know him only by his most wise and excellent contrivances of things”, he is not saying that we know God where we have failed to understand nature. For Newton, the list of “his contrivances” is the list of everything that that exists outside of God. Newton’s comment is explained by the immediately preceding sentences:

We have ideas of his attributes, but what the real substance of anything is, we know not. In bodies we see only their figures and colours, we hear only the sounds, we touch only their outward surfaces, we smell only the smells, and taste the savours; but their inward substances are not to be known, either by our senses, or by any reflex act of our minds; much less then have we any idea of the substance of God.

This is a restatement of a principle shared with Aristotle and the Scholastics: “Nothing is in the intellect that was not first in the senses”. Once again, Newton is directly contradicting Tyson: we infer about God through what we *know and observe* about the world around us, “from the appearances of things”, not in scientific ignorance.

As we noted in Part 1, Newton does discuss areas of scientific ignorance in the Scholium: we know about gravity, but we not know “the cause of its power”. Newton, contra Tyson, does not leap at the opportunity to invoke God, but simply says “I frame no hypotheses [hypotheses non fingo]”.

Later in the article, Tyson says,

Newton feared that all this pulling would render the orbits in the solar system unstable. So Newton, in his greatest work, the

Principia, concludes that God must occasionally step in and make things right:“The six primary Planets are revolv’d about the Sun, in circles concentric with the Sun, and with motions directed towards the same parts, and almost in the same plane.… But it is not to be conceived that mere mechanical causes could give birth to so many regular motions.… This most beautiful System of the Sun, Planets, and Comets, could only proceed from the counsel and dominion of an intelligent and powerful Being.”

Tyson has Newton wrong. This passage is about God’s establishment (“proceed from”) of the created order, not miraculous intervention. As we saw last time, the point about the stability of the Solar System is actually found in *Opticks*, not the *Principia*, and plausibly isn’t about miraculous intervention either.

Tyson says of Ptolemy,

Armed with a description, but no real understanding, of what the planets were doing up there, he could not contain his religious fervor:

“I know that I am mortal by nature, and ephemeral; but when I trace, at my pleasure, the windings to and fro of the heavenly bodies, I no longer touch Earth with my feet: I stand in the presence of Zeus himself and take my fill of ambrosia.”

Once again, we must ask: how does this prove that “They appeal to a higher power only when staring into the ocean of their own ignorance”? Read the quote again; Ptolemy is saying the exact opposite. He says that he can “trace, at my pleasure, the windings to and fro of the heavenly bodies”. In other words, it is what he *knows and understands* about the heavens, as well as his own mortality, that fills him with awe.

Further, he is not appealing to God to explain the motion of the heavens. He feels *awe*, not ignorance. What astronomer, what human being, has not felt wonderment at the sight of the stars? Here’s Bill Watterson:

If the stars should appear one night in a thousand years, how would men believe and adore; and preserve for many generations the remembrance of the city of God which had been shown! But every night come out these envoys of beauty, and light the universe with their admonishing smile.

I am amazed that Tyson doesn’t recognise Ptolemy’s “shuddering before the beautiful“. Have the stars never filled him with ambrosia?

Tyson’s says of one of Christiaan Huygens’s books, The Celestial Worlds

Discover’d:

God is absent from this discussion [of astronomy]. Celestial Worlds also brims with speculations about life in the solar system, and that’s where Huygens raises questions to which he has no answer. That’s where he mentions the biological conundrums Of the day, such as the origin of life’s complexity. And sure enough, because seventeenth-century physics was more advanced than seventeenth-century biology, Huygens invokes the hand of God only when he talks about biology:

“I suppose no body will deny but that there’s somewhat more of Contrivance, somewhat more of Miracle in the production and growth of Plants and Animals than in lifeless heaps of inanimate Bodies. … For the finger of God, and the Wisdom of Divine Providence, is in them much more clearly manifested than in the other.”

The opening claim is false; before Huygens discusses life, he refers to “the greatest part of God’s Creation, that innumerable multitude of Stars”. (All Huygens quotes from here.) He further objects to those who would,

… pretend to appoint how far and no farther Men shall go in their Searches, and to set bounds to other Mens Industry; just as if they had been of the Privy Council of Heaven: as if they knew the Marks that God has plac’d to Knowlege.

Huygens is obviously not content with ignorance, and rebukes those who try to limit human knowledge of the universe. Rather, he praises,

That vigorous Industry, and that piercing Wit were given Men to make advances in the search of Nature, and

there’s no reason to put any stop to such Enquiries[emphasis added].

Further, before discussing life, Huygens notes that through astronomy, by viewing Earth – “this small speck of Dirt” – from on high,

We shall be less apt to admire what this World calls great, shall nobly despise those Trifles the generality of Men set their Affections on, when we know that there are a multitude of such Earths inhabited and adorned as well as our own. And we shall worship and reverence that God the Maker of all these things; we shall admire and adore his Providence and wonderful Wisdom which is displayed and manifested all over the Universe.

As with Newton, Huygens sees God everywhere, not in scientific ignorance. Tyson’s claim that “Where they feel certain about their explanations, however, God gets hardly a mention” is simply false. There are 5 mentions of God by Huygens (out of 18 in Book I), and 2 of the “Almighty” or “Supreme Creator”, before Tyson’s quoted section.

Within this context, Huygens notes that in living things God is “more clearly manifested”. This is not about where God acts, but where God’s action is most *obvious*. He has already stated that God’s wisdom is “displayed and manifested all over the Universe”; for Huygens, life is an exceptional example of this.

Tyson thesis fails. Huygens states exactly the opposite opinion: it is in greater and deeper study of the natural world that he contemplates its Creator. He finds God in knowledge, not ignorance, praising

… the contemplation of the Works of God, and the study of Nature, and the improving those Sciences which may bring us to some knowlege in their Beauty and Variety.

And what difference is there between a Man, who with a careless supine negligence views the Beauty and Use of the Sun, and the fine golden Furniture of the Heaven, and one who with a learned Niceness searches into their Courses; who understands wherein the Fixt Stars, as they are call’d, differ from the Planets, and what is the reason of the regular Vicissitude of the Seasons; who by sound reasoning can measure the magnitude and distance of the Sun and Planets? Or between such a one as admires perhaps the nimble Activity and strange Motions of some Animals, and one that knows their whole Structure, understands the whole Fabrick and Architecture of their Composition? [emphasis added].For without Knowlege what would be Contemplation?

Filed under: Astronomy, Science ]]>

**Abstract:** Neil deGrasse Tyson has argued that Isaac Newton’s religious views stymied his science, preventing him from discovering what Laplace showed a century later – that the planetary orbits are stable against perturbation. This conclusion is highly dubious. Newton did develop perturbation theory, and applied it to the moon’s orbit. His lack of progress is explainable in terms of his inferior geometrical, rather than algebraic, approach. Laplace built on the important work of Clairaut, Euler, d’Alembert and Lagrange, which was not available to Newton. Laplace’s discovery was not definitive – computer simulations have showed that the Solar system is chaotic. And finally, Newton does not give up on science and invoke God at the first sight of ignorance, saying rather “I frame no hypothesis”. His “Reformation” of the Solar System is plausibly not supposed to be miraculous. I conclude that scientists (myself included) are terrible at history.

I’ve got a lot of time for Neil deGrasse Tyson, who is doing a wonderful job of bring the excitement and importance of science to the general public and to the next generation in particular. Despite its seeming disconnect from everyday life, astronomy is an important way to get people into science. Plenty of people who are now solving all manner of important problems in our society got interested in science via astronomy.

(Incidentally, I was a dinosaur nerd first. Put me in a science museum and I’m going straight for the fossils.)

I have a problem, however, with this clip (It’s long; I’ll quote the relevant bits below). In it, Tyson discusses a famous story about a conversation between physicist Pierre-Simon Laplace and Napoléon Bonaparte in 1802. Here is a passage from A Budget of Paradoxes by Augustus De Morgan (1872, p. 249-50), which is the earliest account that I can find.

The following anecdote is well known in Paris, but has never been printed entire. Laplace once went in form to present some edition of his ‘Systeme du Monde’ to the First Consul, or Emperor. Napoleon, whom some wags had told that this book contained no mention of the name of God, and who was fond of putting embarrassing questions, received it with — ‘M. Laplace, they tell me you have written this large book on the system of the universe, and have never even mentioned its Creator.’ Laplace, who, though the most supple of politicians, was as stiff as a martyr on every point of his philosophy or religion … drew himself up, and answered bluntly, ‘Je n’avais pas besoin de cette hypothese-la.’ (I had no need of that hypothesis.) Napoleon, greatly amused, told this reply to Lagrange, who exclaimed, ‘Ah! c’est one belle hypothes ; ca explique beaucoup de choses.’ (Ah, it is a fine hypothesis; it explains many things.)

No eyewitness reports Laplace’s zinger. We don’t get this story from Napoleon, Laplace or Lagrange. The only person in the room whose account we have is the British astronomer William Herschel, and he does not record the story above, but rather notes that “Mons. De la Place wished to shew that a chain of natural causes would account for the construction and preservation of the wonderful system. This the first Consul rather opposed.”

There is even evidence that Laplace opposed this anecdote, demanding its deletion from a forthcoming publication shortly before his death in 1827. (Source. p. 111.)

So the story is at least suspect. Laplace’s agnosticism needed a pithy parable, it seems, and someone obliged. The story should not be told without at least some sort of caveat.

Napoleon was into the physics, the engineering and the material science of war. And so he immediately summoned up the five-volume production of Laplace and read it through, cover to cover. He called in Laplace and – I have the exact quote here – asked him what role God played in the construction and regulation of the heavens. That’s what Newton would ask. Laplace replies ‘Sir, I had no need for that hypothesis.’

A few details to note. Tyson says: “I have the exact quote here”. No, he doesn’t because no one does. The story is at best hearsay.

Even our dubious version of the story from De Morgan has Napoleon being informed about Laplace’s book “by some wags” and saying to Laplace *“they tell me”* that the book doesn’t mention God. So Tyson’s “cover to cover” detail is doubtful.

Note also Tyson’s version of Napoleon’s question. The earliest versions of the story have Napoleon asking about the absence of God from the *book*, not God’s role in the whole scheme of things. This exaggerates the scope of Laplace’s supposed answer. Stephen Hawking appreciates this point: “I don’t think that Laplace was claiming that God does not exist. It’s just that he doesn’t intervene, to break the laws of Science.”

Let’s assume for the moment that the story is at least reflective of *some* conversation between Laplace and Napoleon. I’m particularly interested in the moral that Tyson draws from this episode. Tyson claims that Newton (1642-1727) should have discovered what Laplace (1749-1827) did – that that the combined pull of the planets on each other do not destabilise their orbits – but was hamstrung by his theism.

What concerns me is, even if you’re as brilliant as Newton, you reach a point where you start basking in the majesty of God, and then your discovery stops. It just stops. You’re no good any more for advancing that frontier. You’re waiting for someone to come behind you who doesn’t have God on the brain and who says “that’s a really cool problem, I want to solve it.” And they come in and solve it.

But look at the time delay – this was a hundred-year time delay. And the math that’s in perturbation theory is like crumbs for Newton. He could have come up with that. The guy invented calculus just on a dare, practically. When someone asked him why planets orbit in ellipses and not some other shape, and he couldn’t answer that, he goes home for two months and comes back: out comes integral and differential calculus, because he needed that to answer that question.

This is the kind of mind we’re dealing with in Newton. He could have gone there but he didn’t. His religiosity stopped him.

You should be immediately suspicious of Tyson’s account for this reason: Newton and Laplace weren’t the only two physicists on the face of the planet in the 17th and 18th century. Even if Newton was held back, what’s everyone else’s excuse? Did everyone catch Newton’s God-bothering disease, and only Laplace found the cure?

Hardly. Here’s a few relevant historical details.

Tyson’s “he could have gone there but didn’t because of religion” is immediately derailed by the fact that Newton went there. Here is historian William L. Harper, quoting Newton:

… Newton developed this method in an effort to deal with the extreme complexity of solar system motions. … The passage continues with the following characterization of the extraordinary complexity of these resulting motions.

“By reason of the deviation of the Sun from the center of gravity, the centripetal force does not always tend to that immobile center, and hence the planets neither move exactly in ellipses nor revolve twice in the same orbit. There are as many orbits of a planet as it has revolutions, as in the motion of the Moon, and the orbit of any one planet depends on the combined motion of all the planets, not to mention the action of all these on each other. But to consider simultaneously all these causes of motion and to define these motions by exact laws admitting of easy calculation exceeds, if I am not mistaken, the force of any human mind.” (Wilson 1989b, 253)

It appears that shortly after articulating this daunting complexity problem, Newton was hard at work developing resources for responding to it with successive approximations.The development and applications of perturbation theory, from Newton through Laplace at the turn of the nineteenth century and on through much of the work of Simon Newcomb at the turn of the twentieth, led to successive, increasingly accurate corrections of Keplerian planetary orbital motions. [emphasis added]

Indeed, Newton developed two perturbation methods, one of which “corresponds to the variation of orbital parameters method first developed in 1753 by Euler and afterwards by Lagrange and Laplace.”

Why did Newton not achieve what Laplace did a century later? We have seen that it is not from want of trying. He was primarily interested in calculating the moon’s orbit, which is unavoidably a three-body problem: one cannot meaningfully simplify the problem by considering only the Moon and the Earth. Newton applied his method to the Moon, but not successfully. The first edition of the *Principia* notes: “These computations, however, excessively complicated and clogged with approximations as they are, and insufficiently accurate, we have not seen fit to set out.” Later editions remove this comment entirely. Newton was obviously dissatisfied with his calculation.

Why was Newton’s calculation unsuccessful? Was he too busy “basking in the majesty”? Historians have a more mundane explanation.

The first successful derivation of the Moon’s apsidal motion (or rather, of most of it) was announced some sixty years later, by Alexis-Claude Clairaut, in May 1749. Euler obtained a derivation in good agreement with Clairaut’s by mid-1751. … Jean le Rond d’Alembert published a more perspicuous derivation, with the degree of approximation made explicit, in 1754. Success came for Newton’s successors only with a new approach, different from any he had envisaged: algorithmic and global. The Continental mathematicians began with the differential equation, the bequest of Leibniz.

From Newton to d’Alembert, the essential theoretical advance in lunar theory consisted in the decision to start from a set of differential equations, while relinquishing the demand for direct geometrical insight into the particularities of the lunar motions.

Chris Smeenk and Eric Schliesser (highly recommended!) conclude:

Newton also faced a more general obstacle: within his geometric approach it was not possible to enumerate all of the perturbations at a given level of approximation, as one could later enumerate all of the terms at a given order in an analytic expansion. It was only with a more sophisticated mathematics that astronomers could fully realize the advantages of approaching the complexities of the moon’s motion via a series of approximations.

This is one of the most surprising things to the modern physicist about Newton’s *Principia*: having invented calculus, Newton doesn’t really use it. He thought that geometry was more insightful, more fundamental. This prohibited Newton from developing the analytic tools needed to incorporate the perturbations of the other bodies in the Solar System into his model, and – crucially – to evaluate the accuracy of his approximations. Moreover, Newton’s version of calculus is actually rather clunky compared to Leibniz’s, which was being used on the Continent.

Obviously, if Newton’s approach is floundering on the three-body problem (Sun, Earth, Moon) and a few centuries of observational data, the problem of the stability of all the planets in the Solar System into the indefinite future cannot be attacked with much confidence.

The idea that Newton could have come to the conclusions that Laplace did is extremely doubtful. We have already seen that his methods are not quite up to the task. Further, note the mathematicians who worked on the problem of perturbations to planetary orbits ** before** Laplace: Clairaut, Euler, d’Alembert, and Lagrange. These are the greatest mathematicians of their age; Leonard Euler is arguably the greatest mathematician of all time: “Read Euler, read Euler, he is the master of us all.” That quote, incidentally, is from Laplace. Euler was a devout Christian and a Lutheran Saint. Apparently, having “God on the brain” didn’t prevent him – as it didn’t prevent Newton – from working on this scientific problem.

So, I think we can safely say that if Leonard Euler attempts to solve a mathematical problem and fails, the problem is a difficult one. And he took the problem very seriously: when Clairaut successfully applied perturbation theory to the Moon’s orbit, Euler described “this discovery as the most important and profound which has ever been made in mathematics.”

But these mathematicians didn’t merely make failed attempts; they laid the foundations for Laplace’s work. Joseph-Louis Lagrange, in particular, is crucial:

Jacques Laskar gives Lagrange equal credit, referring to the “Laplace-Lagrange stability of the Solar System“:

Laplace and Lagrange, whose work converged on this point, calculated secular variations, in other words long-term variations in the planets’ semi-major axes under the effects of perturbations by the other planets. Their calculations showed that, up to first order in the masses of the planets, these variations vanish.

Newton, of course, was a mathematical genius. But we can hardly blame him for not being smarter than Clairaut, Euler, d’Alembert, Lagrange and Laplace *combined*.

Did Newton ignore the problem of the stability of the Solar System so that he could call upon God as an explanation? Well, we have already seen that he did not ignore the problem at all.

Further, Newton developed his perturbation methods in 1685-6, according to Michael Nauenberg. The definitive statement of his theological conclusions drawn from physics comes in the General Scholium, an essay appended to the end of the second and third editions of the *Principia* in 1713 and 1726 respectively. With 40 years of reflection on the problem of the perturbations of the orbits of the planets and their theological implications, what does Newton have to say about God’s *intervention* in the universe?

Nothing. Nada. Zilch.

Newton states that “This most beautiful System of the Sun, Planets, and Comets, could only proceed from the counsel and dominion of an intelligent and powerful being.” This is about the creation of the whole physical order in the first place, not about God *intervening* in the laws of nature to perform a miracle: “In him are all things contained and moved; yet neither affects the other: God suffers nothing from the motion of bodies; bodies find no resistance from the omnipresence of God.”

In the Scholium, Newton notes a major gap in scientific knowledge:

Hitherto we have explain’d the phænomena of the heavens and of our sea, by the power of Gravity, but have not yet assign’d the cause of this power.

Here is a golden opportunity for Newton, facing a scientific gap in our knowledge, to invoke God as the very power of gravity itself. His famous response:

But hitherto I have not been able to discover the cause of those properties of gravity from phænomena, and

I frame no hypotheses[hypotheses non fingo]. For whatever is not deduc’d from the phænomena, is to be called an hypothesis; and hypotheses, whether metaphysical or physical, whether of occult qualities or mechanical, have no place in experimental philosophy.

For a fellow who is supposed to be looking for any gap in scientific knowledge as an excuse to insert the direct intervention of God, Newton’s doing it all wrong.

Interestingly, Newton seemingly appeals to the intervention of God in a single sentence in Opticks, published in 1704.

For it became [God] who created them to set them in order. And if he did so, it’s unphilosophical to seek for any other Origin of the World, or to pretend that it might arise out of a Chaos by the mere Laws of Nature; though being once form’d, it may continue by those Laws for many Ages. For while Comets move in very excentrick Orbs in all manner of Positions, blind Fate could never make all the Planets move one and the same way in Orbs concentrick,

some inconsiderable Irregularities excepted, which may have risen from the mutual Actions of Comets and Planets upon one another, and which will be apt to increase, till this System wants a Reformation.

This is a puzzling passage, for a number of reasons. Firstly, it seems out of place – the context is about God’s sustaining of the “wonderful Uniformity in the Planetary System”. Mentioning “inconsiderable Irregularities” somewhat undermines Newton’s point.

Secondly, nowhere in Newton’s corpus can we find the calculations to sustain this claim, even though (as we noted above) he had pioneered perturbation theory. How did Newton convince himself that the mutual attractions of the planets increase the irregularities of the Solar System? We don’t know.

Thirdly, why is this not mentioned in the General Scholium? If this is Newton’s great scientific proof of God’s intervention in the world, why does it not appear in his most famous essay on God, at the conclusion of his scientific *magnum opus*? In light of its absence from the *Principia*, it is difficult to know how much weight Newton placed on this particular argument.

Finally, it is difficult to know exactly what Newton was thinking. It is, after all, a single sentence with no further comment. Leibniz, for example, responded to this passage in November 1715:

According to [Newton’s] Doctrine, God Almighty wants to wind up his Watch from Time to Time: Otherwise it would cease to move. He had not, it seems, sufficient Foresight to make it a perpetual Motion. Nay, the Machine of God’s making, is so imperfect, according to these Gentlemen; that he is obliged to clean it now and then by an extraordinary Concourse. … I hold, that when God works Miracles, he does not do it in order to supply the Wants of Nature, but those of

Grace.

In response to Leibniz, Samuel Clarke argues in 1716 that Newton is not appealing to a miracle or a suspension of the laws of nature:

… the word

correction, oramendment, is to be understood, not with regard to God, but to us only. … But this amendment is only relative, with regard to our conception. In reality, and with regard to God; the present frame, and the consequent disorder, and the following renovation, are all equally parts of the design framed in God’s original perfect idea.

In other words, Newton’s “reformation” would be entirely natural, part of God’s orderly sustaining of the universe, rather than a violation of its laws. (I recommend this article for more details). Newton is not committing the God of the gaps fallacy, because he does not see a gap.

Finally, Laplace and Lagrange’s demonstration of the stability of the Solar System was shown by later scientists to be inconclusive. Henri Poincaré established that it was impossible to produce exact solutions to the equations of motion in the n-body problem, where n is bigger than 2: approximate solutions by means of infinite series are the only viable solutions. Moreover, these series generally diverge, making them useless for prediction over infinite time. Laplace and Lagrange’s calculation is informative but not decisive.

Since Poincaré, computer simulations have shown that the orbits of the Solar System are chaotic over timescales of a few billion years. So the “Laplace solves it” part of Tyson’s story has a problem: Laplace didn’t solve it.

What historians do is read primary sources, in the original languages as much as possible, consider all the characters involved, trying to understand their context, their influences, their personal lives and their professional motivation. Nuance, nuance, nuance.

What amateurs do – myself included – is read secondary sources, skim for interesting sections, pick favourites, judge anachronistically, and hope that an amusing anecdote or two can summarise an entire cultural milieu. (Huxley vs Wilberforce is a great example.) We want simple stories of progress, pithy quotes and heroes who look like us.

The historian Steven Shapin, reviewing Steven Weinberg’s recent book “To Explain the World”, gives the view from his discipline of scientists who attempt to write history.

There’s a story told about a distinguished cardiac surgeon who, about to retire, decided he’d like to take up the history of medicine. He sought out a historian friend and asked her if she had any tips for him. The historian said she’d be happy to help but first asked the surgeon a reciprocal favor: “As it happens, I’m about to retire too, and I’m thinking of taking up heart surgery. Do you have any tips for me?”

To illustrate the point, we can retell our story to make Newton the hero. Inspired by God’s providence, he argued correctly that the Solar System is ultimately unstable and was beautifully vindicated by modern computer simulations. Laplace (cue the villainous music), desperately seeking to avoid God, promotes the idea that the Solar System is perfectly stable, allowing his agnosticism to hold back the progress of science and delay the discovery of long-term chaos among the planets by Poincaré and modern physicists.

We can even make Lagrange the hero, since he is at least as important as Laplace in the scientific study of the Solar System. Lagrange is there with Napoleon – so the story goes – to counter Laplace’s myopic inference that, since God doesn’t poke the planets moment by moment, He is not needed. Shall we immortalise Lagrange’s answer to Napoleon, rather than Laplace’s?

This is whig history – the heroes of the past are the people who, for whatever reason, believe something like what I believe now. Why does Tyson venerate Laplace, the agnostic? Because Tyson is an agnostic. That’s all the story proves.

Neil deGrasse Tyson on Newton (Part 2) is here.

Filed under: Astronomy, Science ]]>

**Title: The Fine-Tuning of the Universe for Intelligent Life**

**Abstract: **Let’s make it slightly different from the one that we are familiar with. We could change the laws of nature, just a little bit. We could change how the universe begins, or make it four-dimensional. In the last 30 years, scientists have discovered something astounding: the vast majority of these changes are disastrous. We end up with a universe containing no galaxies, no stars, no planets, no atoms, no molecules, and most importantly, no intelligent life-forms wondering what went wrong. This fact is called the fine-tuning of the universe for life. After explaining the science of what happens when you change the way our universe works, we will ask: what does all this mean?

Filed under: Uncategorized ]]>