## Bad Science, Worse Philosophy: the Quackery and Logic-Chopping of David Foster's The Philosophical Scientists (2000)

# 5. Why Foster Needs to Take a Basic Statistics Course

## Richard Carrier

Foster's book is entirely dependent upon statistics, and his equations and calculations look impressive. But how he came about the use of them is a serious question, for he betrays an astonishing ignorance of even the most basic principles of statistical science. Worse still, those equations and calculations that he does use are actually totally unsuited to what he wants to accomplish, and thus his results are wholly irrelevant to the conclusion he claims to have drawn from them. This latter point will be made much more clear in a later section, when I address his main arguments in detail. But here it will suffice to show his disturbing ignorance of some basic principles of statistics.

Certainly, we can excuse him for making the ever-common hyperbole on page 81 that "nothing can have happened which would need more time than the life of the universe" when what he really means to say is that something that is immensely improbable cannot be assumed to have happened. For no matter how improbable something is, it does not follow that it is *impossible*. And it may indeed simply be the case that something incredibly improbable has happened. After all, given the immense size of the universe and the complexity of what goes on there, one cannot doubt that the immensely improbable often does occur. But that is a fine point that does not suggest any real error, for if that were his only mistake, his argument would still be sound enough to accept.

But he makes far more profound errors. Having studied statistics myself, I was shocked to discover the strangest of statements beginning on page 39. There, he makes the truly bizarre claim that "There has been great confusion among mathematicians as to whether there really is such a matter as the Law of Probability, i.e. a law which tells you how likely something is to happen. At the present time the experts," whom he does not name or cite, "take the view that it is only legitimate to consider that there may be a Law of Improbability, a law concerning the *unlikelihood* of something happening" (sic). I can say with much certainty that no 'expert' has ever said anything of the sort. For the Law of Complementarity is one of the essential foundation stones of all statistical mathematics (cf. Mario Triola, *Elementary Statistics*, 5th ed., Addison-Wesley, 1993, pp. 146ff.). You cannot have an improbability of an event without a complementary probability. For example, if an event has an improbability of 1 in 5, then it has, as a matter or irrefutable mathematical necessity, a probability of 4 in 5. So here, Foster betrays ignorance of the most fundamental lesson found in the most basic introductory textbooks on statistics. His credibility as someone who 'knows statistics' is thoroughly undermined by this astonishing gaffe. It is all the worse for the fact that he claims the authority of nameless 'experts' for what is in fact an instance of his own *lack* of expertise. Perhaps Foster has merely misunderstood some expert writing on the subject.

This ignorance infects other aspects of Foster's argument. On page 58 he develops a concept of 'specificity,' which he explains is equal to 1/Entropy (cf. p. 39). He says this is "the measure of the improbability of a pattern *which actually occurs* against a background of alternatives" (sic). Where entropy is a measure of the number of possible states of a system (this is a confusion of the facts: see my addendum on Entropy), specificity is the probability of getting one specific state out of all of those. He has already misstated himself by claiming that specificity measures the improbability of one state, when it actually measures the *probability* of that state (as per his own example on page 39). The improbability would not be 1/Entropy, but 1-1/Entropy. Anyone who actually understood statistics would know that: this is the Law of Complementarity, the ignorance of which he has already shown.

Now, on page 58, he claims that "it is known that [entropy] refers to the behaviour of 'crowds' whereas specificity refers to individuals." This is entirely wrong, even given his own definitions. Both describe single systems comprised of a crowd of elements. Entropy (i.e. logical or "informational" entropy, not thermal entropy) merely describes the number of states that a crowd can be in (and still look the same); Specificity describes the odds of that crowd being in one particular state among all possible states at any given time. You may think he is merely being ambiguous, that he really means by 'individual' the 'individual state of the crowd' and by 'crowds' the 'number of possible states of the crowd,' but if so he would not then go on to ask this odd question: "are [logical] entropy and specificity two aspects of the same thing?" Of course they are! But he calls this "a paradox"! It is no paradox. It is simply the mathematical inversion of the same measurement.

Foster's use of this concept of 'specificity' also involves some inexcusable blunders of statistical math. On page 121, for instance, he calculates the "specificity" of the conditions for life on Earth as 1 in 720, but his calculations are inherently flawed by the fact that the variables he multiplies are not independent. He openly declares of these variables that "Each of these is relatively independent from the other," but look at his list of supposedly "independent" variables:

- Energy for photosynthesis
- Thermostatic control
- Soil production by weathering
- Leaching of vital trace elements
- Tides and the littorals
- Winds

But all of these he attributes to the sun (he calls them all "solar factors" and argues at length how the sun causes each one). If they were in fact all solar, then they would all be *dependent* variables, not independent variables. For once you have one of them, the odds of having the rest are 100%. After all, you cannot have some of the effects of a sun and not others. Once you have a sun, all of its effects are present. Yet their mutual dependence would forbid the use of multiplication to derive his figure of 1/720, and a single semester of statistics would have taught him that.

But it goes beyond even that glaring error. For Foster is not correct that all of these are attributable to the sun. Thermostatic control is the product of terrestrial spin, and tides require a moon, since the tidal forces of the sun in the absence of the moon would be far too weak to produce the effects he says are necessary. Of course, tides are not really needed for life to evolve. They help only in producing *land-based* life----and they are not even required for that, but merely helpful. So, since he claims to be calculating the odds of "organic life" appearing on Earth, and not any specific form of that life, he cannot honestly include this variable. But even if he were to amend his claim to determining the odds of "land-based life" he would still not be able to include the factor as a requisite, but would have to factor it in as an aide, since it only increases the odds of such life, but is not required for it.

As for thermostatic control, that depends upon the Earth's spin. But this is so highly probable that the odds of any planet *not* having a spin are astronomically low. In fact, every 'requirement' he lists exists several other places in our solar system. Mars has all of them----at least, it would have tides if it still had bodies of liquid---and Venus has all of them but tides. Scientists generally agree that both worlds had oceans, and that they may have birthed life, but circumstances ensured it did not take. Even Earth is destined in due time to suffer the same fate.

Ultimately, Foster's figure of 1 in 720 is flawed in numerous ways that are inexcusable for any attentive student of statistics, and his result is false. Given only the factors he mentions, and the scant information we have from our own solar system, the odds of life forming on Earth would actually be better than 50%. How so? First, all solar factors have a chance of existing of 1 in 1, given the existence of a sun; second, the chance of having a spin is also 1 in 1, but we will be kind to Foster and say the odds are 2 in 3, based on the fact that only six of the nine planets we know have a spin equal to or faster than Earth's (Earth, Mars, Jupiter, Saturn, Uranus, and Neptune); third, the chance of having a moon can be similarly guessed from the limited data of our own solar system--all planets but two (Mercury and Venus) have moons, giving a probability of 7 in 9. So Foster's equation, based solely on factors he himself has given, should look like this: 1 x 2/3 x 1 x 1 x 7/9 x 1 = 52%. Thus, the odds of life forming on Earth, using Foster's own list of conditions, is better than 50/50. Naturally, there are factors he fails to mention, such as a magnetic field, and others he mentions but excludes, such as having an ocean. And then there is the fact, already mentioned, that the influence of the moon would not act as a necessary condition, but a helping one, and so on. But it should be clear that Foster is either incompetent or for some reason trying to pull the wool over our eyes.

In this one case, there are actually things he left out, and therein lies the most bizarre mistake Foster makes here: he claims to have calculated the odds of "organic life on Earth" but in fact he excludes two factors he says are not attributable to the sun (division into oceans and dry land, and atmospheric protection from meteorites and radiation). Honest and valid statistical arguments require that you at least *try* to account for all necessary variables, or else admit that the results cannot be achieved. Even excusing that, how can he be claiming to calculate the odds of organic life, when all he is calculating are the odds of all (in his opinion) solar effects coexisting on earth? But the odds of all solar effects coexisting on earth are 100% the moment you have a sun. The 52% we calculated was based on the fact that two of Foster's "solar factors" are not in fact solar. So Foster has used logical legerdemain to create an entirely bogus statistic that has no bearing on his argument at all.