I suppose it’s not so remarkable that creationists can’t do mathematics. After all, almost by definition, they don’t understand evolution, so that alone should suggest some sort of cognitive deficit. What surprises me is that even creationists with math or related degrees often have problems with basic mathematics.

I wrote before about Marvin Bittinger, a mathematician who made up an entirely bogus “time principle” to estimate probabilities of events. And about Kirk Durston, who speaks confidently about infinity, but gets nearly everything wrong.

And here’s yet another example: creationist Jonathan Bartlett, who is director of something called the Blyth Institute (which, mysteriously, lists no actual people associated with it, and seems to consist entirely of Jonathan Bartlett himself), has recently published a post about mathematics, in which he makes a number of very dubious assertions. I’ll just mention two.

First, Bartlett calls polynomials the “standard algebraic functions”. This is definitely nonstandard terminology, and not anything a mathematician would say. For mathematicians, an “algebraic function” is one that satisfies the analogue of an algebraic equation. For example, consider the function f(x) defined by f^2 + f + x = 0. The function (-1 + sqrt(1-4x))/2 satisfies this equation, and hence it would be called algebraic.

Second, Bartlett claims that “every calculus student learns a method for writing sine and cosine” in terms of polynomials, even though he also states this is “impossible”. How can one resolve this contradiction? Easy! He explains that “If, however, we allow ourselves an infinite number of polynomial terms, we can indeed write sine and cosine in terms of polynomial functions”.

This reminds me of the old joke about Lincoln: “In discussing the question, he used to liken the case to that of the boy who, when asked how many legs his calf would have if he called its tail a leg, replied, “Five,” to which the prompt response was made that calling the tail a leg would not make it a leg.”

If one allows “an infinite number of polynomial terms”, then the result is not a polynomial! How hard can this be to understand? Such a thing is called a “power series”; it is not the same as a polynomial at all. Mathematicians even use a different notation to distinguish between these. Polynomials over a field F in one variable are written using the symbol F[x]; power series are written as F[[x]].

Moral of the story: don’t learn mathematics from creationists.

P.S. Another example of Bartlett getting basic things wrong is here.

Charly says

True, but to nitpick – in certain linguistic contexts it would. Words are defined by use, so if in a language did not distinguish between legs and tails and had one word for both, and that word were “leg”, then tails would be legs in the sense that all tails are legs, but not all legs are (functionally) tails.

For example slavic languages do not distinguish between fingers and toes, there is only the generic term “digit” for both.

leerudolph says

To be fairer to Bartlett than he deserves, in Abraham Robinson’s Non-Standard Analysis one can have that (to quote his book with that title, p. 147) “the degree and the rank of a polynomial may now be infinite natural numbers” (the “rank” is how many coefficients are non-zero). And Polya, in

Induction and Analogy in Mathematics, sections II.5–6, recalls how Euler in effect treated an expression which has “an infinity of terms, is of ‘infinite degree'”, as if it were a polynomial, to derive a conclusion that “was daring. ‘The method was new and never used yet for such a purpose,’ he wrote ten years later. He saw some objections himself and many objections were raised by his mathematical friends when they recovered from their firs admiring surprise.'” Eventually, “He found a new proof. This proof, although hidden and ingenious, was based on more usual considerations and was accepted as completely rigorous.” Ibelievethat I have read (but can’t find it by Googling; I have Polya on my bookshelf, but my present access to Robinson is limited, via Google Books, and anyway I don’t think what I remember is actuallythere) that, with due care, there is an entirely rigorous proof (within NSA) of the given result of Euler that basically follows Euler’s approach by using polynomials of infinite degree.I doubt that Bartlett knows or cares about any of this.

Adrian Keister says

Can you please point out to me in Bartlett’s article where exactly Bartlett claims that the result of writing sine or cosine in terms of polynomials is itself a polynomial? Writing $$\sin(x)=\sum_{k=0}^{\infty}\left[\frac{(-1)^k}{(2k+1)!}\,x^{2k+1}\right]$$ certainly is writing $\sin(x)$ in terms of polynomials, since each $x^{2k+1}$ qualifies on its own as a polynomial, but I don’t see where Bartlett claims that the result of such a sum is itself a polynomial.

Can I also ask why Bartlett’s creationism has anything at all to do with the quality of his mathematics? Because, on the face of it, your argument for why nobody should learn mathematics from creationists goes like this:

Premise: Bartlett is a terrible mathematician.

Premise: Bartlett is a creationist.

Intermediate Conclusion: Therefore all creationists are terrible mathematicians.

Premise: You should only learn mathematics from good mathematicians.

Conclusion: Therefore, you should not learn mathematics from creationists.

Is this a fair assessment of your argument? If so, I have a problem with it:

The Intermediate Conclusion commits the fallacy of hasty generalization and the fallacy of the illicit major or minor, depending on how you group the premises. Bartlett alone, or even Bartlett plus Durston plus Bittinger, is surely not a representative sample of all creationist mathematicians. No statistician would admit that a sample size of 3 is sufficient for anything. But even if Bartlett is a terrible mathematician, the intermediate conclusion that all creationists are terrible mathematicians is completely unwarranted. In the Intermediate Conclusion, you are talking about all creationists, but neither of the two first premises talks about all creationists (the term is not distributed in the premises). So that’s the fallacy of the illicit major or minor.

As for the “standard algebraic functions”, in the ILATE mnemonic device for integration by parts, the A refers to “algebraic”, and is understood to mean, usually but certainly including, polynomials. And that’s what would come to my mind if someone used the term “algebraic functions”. Functions that express the roots of polynomials, of course, are included in the term as well. Perhaps Bartlett should have written, “It is impossible to write these functions in terms of standard algebraic functions (e.g., polynomial functions).” instead of “It is impossible to write these functions in terms of standard algebraic functions (i.e., polynomial functions).” On the other hand, mathematicians, including myself, sometimes have a tendency to dismiss what someone else says to a non-mathematical audience if it isn’t precise. The problem with precise is that it’s often pedagogically or rhetorically unsound. I’m not saying people should lie in order to persuade, just that not all details are critical for every situation. Even mathematicians know that, as proof details vary considerably depending on audience.

shallit says

Dear Adrian “It is my firm belief that evolution is not science” Keister:

You invent an imaginary argument that you ascribe to me. That’s a fallacy called the “straw man”. Look it up.

Nobody equates polynomials with “algebraic functions”. One class is a strict subset of the other.

Adrian Keister says

The last name is “Keister”, not “Keisler”, please.

The whole point of me summing up your argument is to make sure that you understand that I understand your argument, precisely to avoid strawman, of which I am well aware, having taught logic myself.

You have not done what I asked: if my summing up of your argument is not satisfactory to you, then please indicate how and where it doesn’t describe your argument.

You have not responded at all to my comment about where Bartlett claims that the result of summing up an infinite collection of polynomials is itself a polynomial.

Adrian Keister says

Or if I didn’t make it clear, please let me make it clear now: could you please sum up your argument in a premise/conclusion fashion to your satisfaction?

shallit says

You seem very confused. Not every blog post constitutes an argument. My post is commentary, not a formal deductive argument of any kind. You don’t get to make up an argument and ascribe it to me; that’s rude and unprofessional.

As for your other comment, I didn’t reply because I am unable to teach reading comprehension in a comment. It is completely clear from the context that this is what he was implying.

Adrian Keister says

Well obviously you don’t think you have anything to learn from me, as evidenced by your rude, condescending remarks that are clearly calculated to bring this “conversation” to a close. I wouldn’t be the least bit offended if someone tried to sum up an argument I had made, even if not done perfectly. I certainly did not try to “make up an argument” and ascribe it to you. I tried to sum up what I thought your argument was. I certainly don’t regard such doings as rude and unprofessional; it happens all the time as part of scholarship. Your blog post is definitely an argument, because there is a statement at the end, “Moral of the story: don’t learn mathematics from creationists.” which is something you allege to follow from the statements before.

I firmly believe I can learn something from every person on this planet, including yourself; but if you’re not interested, there’s really nothing more to be said.

shallit says

Indeed, you did precisely what I said you did. You made up an argument and ascribed it to me. That’s where the rudeness comes from. You’re just mad that I called you on it.

As for condescending, if you say stuff like “It is my firm belief that evolution is not science”, don’t expect most educated people to take you seriously.

If you think the phrase ‘the moral of the story’ signifies a logical argument has been made, you must be constantly disappointed when you read Christian literature.

shallit says

You are contradicting yourself.

On the one hand, you say he should have said “It is impossible to write these functions in terms of standard algebraic functions (e.g., polynomial functions).” Presumably you intend this to be a true statement.

But you also say “Writing $$\sin(x)=\sum_{k=0}^{\infty}\left[\frac{(-1)^k}{(2k+1)!}\,x^{2k+1}\right]$$ certainly is writing $\sin(x)$ in terms of polynomials”. Presumably you intend this to be true, also.

So you contradict yourself.

Everyone can see the linguistic con you’re running here. The first “in terms of” means a finite expression. In the second you suddenly allow an infinite expression.

Adrian Keister says

My comment was more intended to address the “standard algebraic functions (i.e., polynomial functions)” bit instead of the statement as a whole. As in, comparing the set of algebraic functions to the set of polynomial functions. I’m not sure I agree with the statement “It is impossible to write these functions in terms of standard algebraic functions (e.g., polynomial functions).” as it stands. The expression I wrote out, the standard Taylor series, is writing the sine function in terms of polynomials. I think most mathematicians would agree with that.

So I think you’re right that Bartlett might be over-stating the impossibility of writing sine and cosine “in terms of” polynomials. But also keep in mind his audience, which is clearly not mathematicians. He’s writing for the general public, and possibly students as well, where the subtleties involved might not be appropriate.

shallit says

No, I don’t think “most mathematicians” would agree that a power series expresses “sine and cosine in terms of polynomials”.

I certainly wouldn’t, any more than I would say that a series like π = 3 + 1/10 + 4/100 + 1/1000 + … writes “π in terms of rational numbers”.

Adrian Keister says

I’m still not seeing any necessary or sufficient connection at all between being a creationist and being good (or not) at mathematics, which is the subject of the post. What has the one to do with the other?

Could you just as easily find evolutionists that are bad at mathematics? As there are loads of people bad at mathematics, I would be shocked if there weren’t evolutionists that are bad at mathematics. Does that mean I shouldn’t learn mathematics from an evolutionist? I don’t think so! One of the best math professors I had at Virginia Tech was an evolutionist, and I learned much from him.

shallit says

Well, this one’s easy to answer: people who are ignorant, misinformed, or intellectually dishonest, in one area (evolution) are likely to be ignorant, misinformed, or intellectually dishonest in another.

Your example of the “evolutionist” is not symmetric, since the “evolutionist” is not the one denying established science.

Adrian Keister says

It’s certainly possible that people who are ignorant, misinformed, etc., in one field are more likely to be so in another. But it’s often just the reverse. Exceptional actors, e.g., can (and very often do) have horrible economics and politics. There are enough counterexamples to your claim to make it rather empty. Besides, you seem to equate being against established science with being ignorant, misinformed, and intellectually dishonest. By that reasoning, Isaac Newton would be ignorant, misinformed, and intellectually dishonest for disagreeing with the established Aristotelian physics which had held sway for hundreds of years before him.

You’re painting with too broad a brush, and it just doesn’t hold.

However, I do not sense that there are any words whatsoever I could say that will change your mind. I have no ethos with you, and me spending more time on it is a waste of both our time. It’s your post, so you get the last say, but I’m not inclined to respond further.

shallit says

Taking issue with the results in a paper or two is one thing. Claiming that an entire field with literally thousands of people working and publishing good papers in it, for over 100 years, “is not science” is just deranged.

Adrian Keister says

I use the phrase “in terms of” in its usual sense. If you can write a true equation with $y$ on the LHS (typically), and an expression involving $x$ on the RHS, then you’ve written $y$ ‘in terms of’ $x.$ Certainly, in some contexts, additional clarification is necessary, like saying, “We write it formally as…” to indicate that you’re not saying the equation is true, yet.

We might have to agree to disagree about whether most mathematicians would agree that a power series expresses “sine and cosine in terms of polynomials”. I certainly would say that writing $\pi=3+1/10+4/100+\dots$ is writing $\pi$ in terms of rational numbers. Properties of the parts don’t necessarily translate to the whole, and vice versa. If that’s understood, there’s nothing wrong with the phrase “writing $\pi$ in terms of rational numbers”.

I think the main point of the article is flawed. Even if you find Bartlett’s mathematics wrong, you have no basis whatever for making the immense non sequitur of saying “don’t learn mathematics from creationists.”

shallit says

If you are asked, for example, to write ζ(3) “in terms of” known constants, and your answer is an infinite series of rational numbers,

no onewill take you seriously. I’m beginning to think you’re just trolling.bugzpodder says

shallit, I took some of your theory classes back in the day (ten years ago) and they are absolutely some of the best. Your topics were always explained in a manner that’s crisp clear and are highly enjoyable. Just want to say thank you! It’s always satisfying to see concepts like Thue-Morse sequence pop up from time to time, I can’t believe I’ve never heard of it before you showed it to me!

shallit says

You are too kind. Thank you.

WMDKitty -- Survivor says

Well, when you believe that 1+1+1 somehow equals 1…

shallit says

Yeah, the doctrine of the Trinity is one of the more incoherent & unbelievable aspects of Christianity. Why do people nod and accept such drivel with eagerness?