“If God does not exist, the applicability of mathematics is just a happy coincidence.
But the applicability of mathematics is not just a happy coincidence. Therefore, God exists.
Eugene Wigner was right; the effectiveness of mathematics in the physical world is quite
literally a miracle, which is best explained by the existence of God.” … Or is it?
In fact, are we even justified in considering mathematics to be miraculous? And what’s
more, is the only other option to God really just happy coincidence? These are the primary
questions we’ll delve into within this response to Dr. William Lane Craig’s Mathematical
Argument for the Existence of God. I’m Stephen Woodford, and thanks for joining me. Let’s get to it.
“Why does mathematics work?” Well, that depends on what you mean by Mathematics,
and no, I am not pulling a Jordan Peterson on you here. The debate over what exactly
mathematics is (that is, whether it’s invented or discovered) has raged since the time to
Pythagoras, and as it stands there are many competing schools of thought, which differ
greatly in how they would answer the question of why mathematics works. This, in fact, is
my first objection to Craig’s presentation. Despite the fact that he is well-versed in
mathematics, having studied Cantorian Set Theory for his work on the Kalam, he has not
provided us with a clear definition of mathematics, and so we’re going to have to infer his
definition from his surrounding words. To be clear, this is not an objection to the
argument, but rather to its presentation. In philosophy, definitions are paramount,
especially when the definition of a keyword is contentious. “Think about it. Mathematical
entities like numbers, sets, and equations are non-physical and abstract. They can't
cause anything, yet for some reason, the physical universe operates mathematically.” Mathematical
entities such as the aforementioned are non-physical and abstract, yes, but the things they were
invented to refer to are not. The earliest signs of homo-sapiens exercising mathematics
dates back at least 5,000 years to the Kingdoms of Mesopotamia and Ancient Egypt, in which
arithmetic, algebra and geometry were employed for the purposes of trade, commerce, astronomy
and taxation. As a proponent of the non-platonic view of mathematics known as Logicism, I would
thus argue that mathematics is an expression of logic that holds only so much as the its
axioms correlate with reality. To answer the opening question through this prism: Mathematics
works because that is precisely what we’ve designed it to do – so much so, in fact,
that when it doesn’t work (which is to say, when it doesn’t correlate with reality),
we alter our axioms so that that it does. To ask ‘Why is mathematics so unreasonably
effective?’ is to ask ‘Why is reason so unreasonably effective?’ “As Galileo put
it, “The book of nature is written in the language of mathematics.” “Scientists
do not use mathematics merely as a convenient way of organizing the data, they believe that
mathematical relationships reflects real aspects of the physical world.” As already mentioned,
mathematicians are far from united in their views of what mathematics is, let alone what
its relationship is to the physical world. To build his narrative of mathematics being
miraculous, Craig is careful as to only quote mathematicians who share his view, and in
doing so his presentation heavily gives the impression that mathematics is discovered,
as opposed to invented. To bolster this point, here’s Paul Davies, the physicist that Craig
has just quoted, owning the fact that his views are a minority among the scientific
community. "And in that view you are distinctly in the minority among at least the physicists
and mathematicians and scientists that I know." "Right" "And you don't deny that you're in
the minority of that thinking… a place that you like to inhabit.” “Yes, I freely admit
that most of my colleagues would not agree with me.” Whereas the last objection was
purely of Craig’s presentation, this objection is against both his presentation and argument.
Craig spends the bulk of his presentation selling the false impression that scientists,
mathematicians, and physicists can’t provide a naturalist account for mathematics, which
it vital for his argument to work. If mathematics is not microvillous, his argument is baffling.
“Pythagoras discovered that when a vibrating string is shortened by half it plays the same
note one octave higher.” From here, Craig lists many of humanity’s most astonishing
discoveries, from Newtons laws, to Maxwell’s radio waves, to Einstein’s general relativity,
all while emphasising their eloquent mathematical expressions, and then he asks the big question:
“How do we explain the astonishing applicability of math to the physical world? In 1960, the
Nobel prize-winning physicist and mathematician, Eugene Wigner, published an article that stunned
the scientific community entitled "The Unreasonable Effectiveness of Mathematics in the Natural
Sciences." Wigner concluded that the effectiveness of mathematics is a "miracle... which we neither
understand nor deserve." Wigner’s article was, and indeed still is, contentious among
mathematicians, largely due to there being so many schools of thought, but it’s my
opinion that Craig, in his theological fervour, does Wigner two disservices. The first is
that he misappropriates Wigner’s words to fit the narrative of mathematics being a ‘miracle’
in the divine sense of the word, as opposed to the ‘extraordinarily welcome’ sense
of the word. Within his short article, Wigner uses the word ‘miracle’ twelve times,
as he considers, for instance, the fact that we can predict anything in the future to be
a ‘miracle’, and he compares the ‘miracle’ of mathematics playing a role in quantum theory
to the ‘miracle’ that “The human mind can string a thousand arguments together without
getting into contradictions." And the second disservice I believe Craig does to Wigner
is that by so misappropriating the his words, Craig gives the impression that Winger was
a theist, when in fact, he was an atheist. Craig states that: “Wigner concluded that
the effectiveness of mathematics is a "miracle... which we neither understand nor deserve."
But in order to present this interpretation Craig has had to fuse the title of Wigner’s
paper with various, disjointed parts of his concluding paragraph. For what it’s worth,
here’s how Wigner actually concluded: “The miracle of the appropriateness of the language
of mathematics for the formulation of the laws of physics is a wonderful gift which
we neither understand nor deserve. We should be grateful for it and hope that it will remain
valid in future research and that it will extend, for better or for worse, to our pleasure,
even though perhaps also to our bafflement, to wide branches of learning.” "“Why is
mathematics so effective?” It’s at this point that Craig moves on, but I’d like
to plant a clear flag here before we do this same. Craig has not demonstrated that the
effectiveness of mathematics is miraculous. The reason that mathematics is so unreasonably
effective is the same reason that reason is so unreasonably effective; it’s because
1) when it’s not effective, we alter its axioms so that it is, and 2) the universe
is fairly consistent, and so it’s no surprise that the tools that we create for our own
environment happen to be applicable to other environments. “Why is mathematics so effective?
Philosophers who address this question fall into two camps: naturalists, who believe that
all that exists concretely is space-time and its physical contents –they exclude supernatural
causes – and theists, who believe in a God who created the universe.” The definition
of naturalism, like mathematics, is hotly debated, and the one that Craig presents is
as uncharitable as to define Christians as those who believe in creationism and a young
earth. A more charitable (and indeed accurate) definition of a naturalist is one who believes
that the laws and forces of the universe are governed entirely by natural (as opposed to
supernatural) regulations. Predictive mathematics, ironically enough, depends on naturalism;
our predictions are predicated on the consistency of the laws of nature, and a supernatural
intervention would violate our predictions, since we would not have accounted for it.
“Naturalists cannot provide a reasonable explanation for why mathematics applies to
the physical world. It's just a happy coincidence. But this is no explanation at all. At most,
naturalists can say that it's not surprising that math applies to the world because the
world itself just happens to have a mathematical structure, so of course mathematics applies
to it. But this explanation is unsatisfactory for two reasons.” This is quite the strawman.
There are many ways in which naturalists can account for why mathematics applies to the
physical world. I’ve already given you my own (and I’ll do so again a little later),
but for now, here’s are just three alternative accounts from, in this order, theoretical
physicist Sabine Hossenfelder, theoretical physicist Steven Weinberg, and then cognitive
linguist and philosopher George Lakoof: “But maybe you do not agree that the effectiveness
of mathematics is all that unreasonable?” “No, I do not think that it is all that
unreasonable that mathematics is effective in the Natural Sciences because, what is mathematics
about? It is a way to describe patterns, to describe regularities, and that's exactly
what we do in the Natural Sciences.” “First of all, how accurate is mathematics in explaining
physics?" "I don't think mathematics can be ever regarded as an explanation in itself
of anything; this has not always been well understand, perhaps it’s still controversial.
Physical theories aren’t the way they are because of principles of mathematics, principles
of mathematics are the language in which we state our physical principles, and they are
the way-- the intellectual tools we use for calculating the consequences of those principles,
but nothing is the way it is because of some mathematical principles.” “There is still
a mathematics in the world even if human beings never exists, or brains never existed.”
“It’s not in the world, the world is as it is. Let’s take a very simple case. Take
a spiral nebula […] The logarithmic spiral is not in the nebula, it's in your understanding
of the nebula. The marvellous thing about mathematics is that we can create mathematics
with our brains that fit phenomena in the world remarkably. It is not a miracle that
that's the case because we are have the capacity to see and understand the world to categorize
it in terms of what our brains do, and then we can create a mathematics out of that in
a systemic way using what our brains allow us.” In case you want to hear more from
voices such as those just shown, know that all three of the prior clips are from Robert
Lawrence Kuhn’s excellent show, Closer to Truth, which I’ve linked below for your
convenience. Okay, back to Craig’s argument. “First, a great deal of mathematics in science
cannot be physically realized, for example imaginary numbers and infinite dimensional
spaces.” Indeed, much of mathematics can’t be physically realised, and it’s for this
reason that while I recognise mathematics to be one of humanity’s greatest tools,
I do not assert that all that we infer from its axioms necessarily exist in reality. To
quote Einstein, “As far as the laws of mathematics refer to reality, they are not certain; and
as far as they are certain, they do not refer to reality.” “And second, this answer
still doesn't explain why the physical universe has such a stunningly elegant mathematical
structure.” At the risk of sounding like a broken record, it’s because the laws of
the universe are robustly consistent, and thus the mathematics we developed to account
for our own, macroscopic physics, is applicable to physical realms far beyond what we inhabit.
This is not strange or mysterious, let alone miraculous. It’s expected. “By contrast,
for theists, mathematics works so well in the physical world because God has chosen
to create the world according to the plan he had in mind.” Right, so according to
Craig, the answer “Mathematics is effective because the universe is relatively consistent”
is unsatisfactory, but the answer “Mathematics is effective because of God” is somehow
perfectly satisfactory? God is a non-answer. It’s the equivalent of “because I said
so”. It not only doesn’t answer the ‘how’ element intrinsic to the ‘why’ question,
but it actively shuts down investigation. It stalls progress, and rots the mind. When
Newton’s laws did not account for the trajectory for all the celestial bodies in the solar
system, he invoked God, and in doing so turned his brilliant mind off. So far as he was concerned,
he had answered the question, but he had not. There was a natural explanation, that not
only answered “why”, but more crucially, answered “how” – and the answer, funny
enough, required us to alter our axioms. “Mathematics and physics work so well together because
the same mind that designed the universe on a mathematical model, also built the universe
on the same mathematical model. All of this adds up to an argument for the existence of
God that goes like this. If God does not exist, the applicability of mathematics is just a
happy coincidence.” If you, like me, are not convinced of the proposition that the
effectiveness of mathematics is miraculous in the religious sense, then this premise
will strike you, as it does me, as bewildering. Charles Darwin once wrote that “A mathematician
is a blind man in a dark room looking for a black cat which isn't there”, and I would
say that this fairly summarises the fact that while we have good reason to suspect that
the mathematics that applies to our environment also applies to radically different environments,
we do not know for certain that it does. In any case, I, and evidently most mathematicians
and physicists, do not consider the effectiveness of mathematics to be ‘unreasonable.’ I
would in fact be more shocked, and find it more unreasonable, if our mathematics did
not apply to other other macroscopic environments. This is to say that so far as I’m concerned,
Craig is committing a Black & White fallacy, in which black is God, and white is coincidence
(or chance). He spend a total of 30 seconds attempting to justify why the only other option
to God is happy coincidence, but as expressed earlier, naturalists have a plethora of ways
to account for the effectiveness of mathematics, including the one I have given throughout.
“But the applicability of mathematics is not just a happy coincidence. Therefore, God
exists.” Dr. Craig is very often presented as a theist of strong intellectual stature,
and I have to say, I’m someone who shares this notion. Craig has challenged me, in fact,
more than any other living theist I’ve read, but his argumentation here strikes me as jarringly
poor. By assuming that the properties of the universe are built upon mathematics, rather
than that mathematics is built upon the properties of the universe, Craig has put the cart before
the horse. His argument is essentially a reskin of the presupositionalism, in which he’s
replaced the laws of logic with the axioms of mathematics. Now, to ensure that Craig’s
specific presentation here isn’t simply a weak rendition, I’ve listened to him debating
the philosopher Dr. Graham Oppy on this very argument, and I’ve digested just about every
video on YouTube I can find of him discussing the topic, but I’ve found nothing of greater
substance than what’s found in this polished short presentation. If I’m honest, this
is the kind of argument I’d expect from the likes of Frank Turek, as opposed to Craig.
Because of this, I can’t knock the feeling that I’m missing something. So, if any of
you think that I am, please let me know, and I’ll respond to it in a subsequent video.
Anyhow, before signing off, I have some pretty huge news! Since the beginning of this channel,
I have been filming in my tiny 1 bed flat, and have had to rearrange all the furniture
and reset all the lighting, cameras and audio each and every time I’ve wanted to film,
but as of three weeks time, I’ll be renting an office! The result, I hope, will be a significant
increase in production quality and quantity. And so I just want to thank you all for making
this possible, and, of course, I’d like say a little extra special thank you to my
wonderful patrons and those of you who’ve supported the channel via other means.