Does Applicability of Math Point to Theism or Not?

I’m going to disagree with this.

I disagree with the broad idea that theism is simpler. But I most strongly disagree with the view that the applicability of mathematics is a brute fact.

I actually wrote a blog post about this:
The reasonable effectiveness of mathematics in the natural sciences.


Good topic. Deserves its own thread.

Applicability of math is more than we “named things with numbers.” The amazing thing is manipulating these numerical properties correlates to a surprising degree with the physical world. It is essentially a kind of magic. We write arcane symbolic spells from our minds that bring about desired physical effects.


I suppose for me it is hard to imagine a world where mathematics did not apply. Math is based on very rigorous use of logic, in particular way. Even numbers are defined in group theory as logical constructs.

We exist. It seems that existence implies mathematics works in existence. The question of the thread seems to suggest we could have found an existence were math did not work. I’m hard pressed to imagine this counterfactual existence. Therefore I am not sure what “math” actually adds to the argument. It seems to be merely restating the brute fact:

We exist, and have not good acccout for why we exist.

…without adding to this brute fact.

What am I missing @Philosurfer?

Great topic, not discussed enough I think.

There are a few different aspects I think, and I’m not enough of a mathematician, physicist, or philosopher to really do them justice.

Your blog Neil seems to address the issue of why it is possible to describe the world in abstract terms. Your claim is that the abstract terms (maths) are fictions so all the real work is being done by the underlying physical reality. Why that reality should cohere with the relatively simple abstract pattern though is not so clear - perhaps we can push the problem away with an appeal to something like laws of nature, but a theistic basis for laws of nature is well argued by a range of thinkers, from different angles (Richard Swinburne, Nancy Cartwright [atheist - denies laws], John Foster, etc.) A nice example of one aspect that seems surprising in the mapping between maths and physics is that gigabytes of experimental data at CERN can be described with just a few equations - it could have conceivably been very different.

A distinct issue from ‘why maths works at all’ and ‘why there is a relatively simple mathematical order to the unvierse’ is why our invented abstract terms work so well so often. Why our brains evolved for survival on the savannah seem to give us insight into quantum mechanics etc. We seem to use aesthetic judgments when developing advanced mathematical physics - the connection there is quite mysterious.

This is a good debate on the topic - the agnostic Daniel Came does a better job than William Lane Craig of spelling out some of the difficulties for naturalism, as I saw it …


Have you ever thought about a town where the streets are laid out in a grid and numbered. So we have 1st street, 2nd street, etc. And, orthogonal to those we have 1st avenue, 2nd avenue, etc. It is very easy to use mathematics with street addresses.

If, however, we had named the streets: 3rd street, 7th street, 2nd street – in a messed up order, and a similar mixed way of naming the avenues, the mathematics wouldn’t work.

The mathematics correlates with the physical world, because we carefully chose a naming convention that would impose that correlation on the world.

I disagree with that. Geometry is also part of mathematics.

Logic works with abstract symbols. Geometry (measuring the world) works with a world and designs way of attaching symbols to that world.

The two, geometry and logic, complement one another.

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Isn’t it amazing the physical world is so consistent, instead of like your messed up streets example? This is the reason mathematical reasoning works so well.

Sure. That’s fine. Still…

Can you show me what an existence in which mathematics did not work would look like?

Take a look at an orchard, with it carefully arranged rows of fruit trees. And then look at a natural forest. The natural forest is more like the messed up streets.

I always tell people that mathematics is not about reality. It is about our behavior. Arithmetic is an idealization of our counting behavior. Geometry is an idealization of our measuring behavior.

As long as we are in a world where we can exist, we will have behavior that we can idealize. So the mathematics will always be useful. It is useful to compare idealized behavior with actual behavior, because the difference between the two is empirically significant.

But we are not labelling the messy bits with science. We label the well ordered structure. That’s why science works.

Mathematics works best in physics. And a lot of physics deals with abstractions (time, distance, etc). The scientists create those ordered structures. And it is because the scientists create them, that they are able to ensure that they are ordered.

Interesting. I did not know the scientists created space and time. Do you know who I can pay to get more time?

They (or someone) created orderly ways of measuring distance and time. And then they abstracted from that measuring system.

That’s different from scientists creating time and space. You are saying scientists invented rulers and clocks. Quite true, and doesn’t make it less amazing that rulers and clocks measure certain quantities so consistently throughout the entire universe.

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Yes. But they extrapolate to way beyond where (or when) we could ever measure. And that amounts to creating space and time as an abstraction.

You are unable to see it, because it is so well hidden. But it is hidden in plain sight.

Here’s a way of measuring short distances. Take a long piece of string, and tie a series of knots in it. Then dab some paint on each knot, using different colors. To measure a distance, you stretch out the string, find the nearest knot and use its color to name the distance.

That would work quite well. You could use it to compare distances, as long as you always use the same piece of string.

The mathematical way: Make those knots exactly equally spaced, and number them sequentially. Then you can use mathematics. That’s how we build mathematical applicability into our concepts.

To what extent is selection relevant here? I.e. Math is not applicable to all aspects of the universe, and physics is the study of those aspects of the universe that happen to be well described mathematically. Thus math seems remarkably well suited because we’ve already filtered out all the problems for which it isn’t.

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Neil, I think you never really responded adequately to the example I brought up in the thread about Newtonian mechanics: the g-factor of the electron. Your argument only works in the cases where physics didn’t match up well with reality, and theorists added an additional term or concept to account for that discrepancy. But the g-factor is far more special. It is computed from quantum electrodynamics (QED), which is a theory unifying special relativity and quantum mechanics to describe electromagnetism. So, the ingredients for this theory were already present in the early 20th century. And sure, there are some parts of the QED (such as renormalization) which you can chalk up to “physicists inventing new concepts to redefine reality”. And it took some time for physicists to develop QED. But apart from that, the theory was basically finished by 1948. At that point, Schwinger and Feynman had only calculated the g-factor to be

g_e/2 = 1.00116,

The calculation for this is (relatively) simple and elegant. We are taught it in beginner quantum field theory class, such that even a mathematically limited experimentalist like me can redo it.

Fast-forward to 2008. A group of Japanese theorists had calculated g_e with the same physics as Feynman did in 1948, but with much more computing power, allowing them to calculate more terms in the infinite series of Feynman diagrams. They obtained

g_e/2 = 1.001 159 652 181 88 (78).

(The number in brackets is the uncertainty). Then, the electron g-factor was measured experimentally with incredible precision a few meters away from my office, by trapping a single electron in a Penning trap. (The physics of this experiment is also breathtakingly beautiful - they basically created a giant artificial atom that the single electron “orbited” around.) The result was:

g_e/2 = 1.001 159 652 180 85 (76)

In other words, physics that was “invented” to explain the g-factor to 6 decimal places, was found 60 years later to be accurate to 12 decimal places! I believe that this a prime example of the surprising ability of mathematical methods in physics to predict the natural world. Nobody needed to add additional terms or concepts to get from 6 to 12 decimals. I don’t know if it points to theism, but I find it difficult to explain this away as simply “physicists giving the right names to parts of nature.” Yes, they did that - but they did so using fairly simple, elegant mathematics, and they were immensely successful at that. I could imagine a world where we could only predict g_e to within an order of magnitude, even with centuries of trying. And sure, not all of physics is as precise as this. But even the existence of a single such case is close to miraculous for me.


I don’t know QED well enough to be able to comment on the specific details.

I think you are perhaps misunderstanding my point. I’m not attempting to cast doubt on physics. Rather, I am questioning the standard philosophical understanding of physics.

When I say that laws are invented, I am not saying that we are inventing reality. Rather, I am saying that the scientists are inventing ways of talking about reality. They are inventing ways of connecting our language expression with reality itself. I am not saying that they are inventing laws that relate existing concepts. Rather, I am saying that laws and concepts go together as a package.

What you describe there would be inventing laws that relate existing concepts. And that is not at all what I am suggesting.

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