Much has been written on this subject, but since science is about demonstration, not just analysis, here is an experiment to help you decide for yourself.

The issue is perhaps best stated as a common question: Why does it work to use math to describe the physical world? Should we go to extremes, such as Carl Sagan (who popularized the notion “God is a mathematician”) or Max Tegmark (who argues that “The Universe ‘is’ mathematics”)?

On the other hand, I have heard engineers literally gasp at the very mention of such ideas.

So what is that all about?

Various authors have their own further agendas in making these claims, of course. But basically the argument is over whether or not the logic of mathematics precedes its instantiation in reality. The argument is made that the math comes first, and then physical reality forms up in conformity with the math. Reality is said to be, not the physical features that we see and touch—at least it is not that ultimately—but rather, ultimate reality is the math behind it all.

So is there a way to put that to the test?

The following is a modification of the very first lab experiment I ever did as a student in college physics, which not incidentally addresses this issue.

Start by making an inclined plane by laying one book flat on a table and leaning a second book against the first so that its bottom is on the table while its top overlies the first book. Place the books at the edge of the table so that, when you roll a marble down the inclined book, the marble will go flying onto the floor. Then with a pencil mark an X near the top of the inclined book. The X will be the place where you always release the marble to let it roll.

That is all there is to the setup.

Now do the experiment. Place the marble at the X and let it roll down the incline.  The marble will go flying off the table onto the floor, so mark the floor with an X where it lands.

But mow (since this is science) do it again, and again. Each time mark the floor with an X where it lands.

If the marble is following a mathematical law, you might start out by expecting it should land at the same place on the floor every time, in obedience to the law.. Right? The marble is just doing what the math says it should do.

But it doesn’t. So do it over and over. Try very hard to always release the marble at exactly the same place.

But you almost always get a different result. (Occasionally, you might get the same result by accident). But you can see for yourself that the world does not act mathematically perfectly every time.

Now, the point is that there are two different ways of interpreting these results. One way is that of Sagan and Tegmark (who are following Plato), and the second way is that of the groaning scientists and engineers (who will tell you that you aren’t really done with the experiment).

Plato’s view is that the world is trying its best to be perfect, but (darn it) it just can’t quite pull it off because of the existence of error in the world. Maybe you aren’t releasing the marble in the exact same manner every time, or perhaps your breathing is disrobing the air differently every time. But the marble is still trying (it is said) to act exactly as per the math.

To see the other interpretation, we have to finish the experiment. That will require a ruler and some paper onto which to record the measurements. (Science always measures).

Find the center of all the X’s on the floor where the marble has landed. (For our purposes, it is good enough just to eyeball it to find where the center must be). Mark it with a filled-in circle.

Then with the ruler measure how far it is from this circle (the center of the X’s) to each X, and write that distance on the paper.

And with that you have finished the experiment. But what have you accomplished?

Two things. Firstly, you have made it so that you can now use math to describe the travels of the marble. Math only works to describe events exhibiting mathematical regularity, and ending up in different places on the floor does not qualify. But if you now pretend that the marble always ends up at the filled-in circle, you can make a formula describing that regularity.

It is called “idealization.” Although there are many ways of doing it, the idea is that, if the real world is not regular enough to use the math, then change how you look at the world so that it does have this regularity. That is done because there is great utility in having the formula. For instance, you could move your books to a different section of the table and use the formula to predict the center of where the marble will land on the floor at this new location.

And idealization is more practical and used more extensively than you might think at first. As long as you have some marbles, let a large number of them scatter randomly onto the floor. If by some bizarre accident they happen to have created what might seem to be a pattern, then with your hand scatter them further. For purposes of this demonstration, the idea is to achieve a state of “no patterns in the marbles” being allowed to remain.

Now, with your mind, look at the marbles and imagine how you would draw “the best line” that you could through the marbles. Make it so that this line is better than any other line at being most in the middle of the marbles (so that the number of marbles is the same on each side of the line). If you want to, you can use a ribbon to help you literally place and see this line.

You have just idealized your randomly scattered marbles because you could write an equation describing the regularity in this line.

But now let’s look at the second thing that you accomplished with this experiment, which is when you measured the distance from each X on the floor to the filled-in circle at their center. What do you gain from knowing these measurements?

Well, let’s suppose that instead of having the marbles run all over the place when they land on the floor, we want to collect them. We could have them land in a basket. But then the idea is to know how big to have the basket be in order to have the marbles always go into it. And we can know that by looking at the measurements.

In other words, science is often about “making something” out of what we have done—it is about “putting together” things like inclined planes and baskets to make a “gismo”—or it is to see how nature has already made something itself. It is not just about the math alone but about using math to put stuff together.

So see what you have done with your experiment. You have made it so that now you can set up your inclined plane made of two books anywhere that you want, and you will know both where to place the basket (centered over the filled-in circle as determined from the math formula) and also how big the basket should be (from your measurements). Or if you should happen to see something similar in nature, you can model it accordingly.

And thus works science and engineering. That is how, say, a plane can fly with reliability. It is not that every part of the plane acts perfectly as per a math equation (as if our marble always hits the filled-in circle) but that the engineers have built into the plane various tolerances, like our baskets, so that they can use idealized equations but still plan for the variation that occurs in reality.

It is because we are fitting things together to make other things that we are able to combine the idealized math with the tolerances. It is not that the math in and of itself correctly predicts where the marble will land. The math in isolation—before we put stuff together—does not have the predictive value used in science to describe the real world.

So now let’s say that you are visiting an engineer as he or she is designing things with tolerances (the baskets) to handle the discrepancies in what the math predicts, and someone says, “God is a mathematician.” Now you know why you you might witness some (shall we say) good-natured lifting of the eyebrows. What really exists in the Universe is all the variations in where the marble lands, not where the math on its own says it will land.

What science achieves by combining tolerances with idealization is the ability to deal with all that variation.

But surely, it might be argued, science is not really about “just pretending”—idealization—such as pretending that the marble always hits the filled-in circle just to be able to write a formula to that effect.

No, I didn’t say that science is about just pretending. It is about measuring, such as finding how far off the marble lands from the ideal. If we wanted to, we could take the length of how far off a marble lands, and divide that by how far the marble travels, to find its “percent error.” And then we could use that in our calculations.

Scientists and engineers are always measuring and always saying how far off their measurements might be. That is not the same as pretending, even when what they are measuring is how far off they are from an ideal.

So now you are in a position to stake out some of your own answers to the questions posed at the top of this essay. Consider your experiment and answer: Is mathematics what is ultimately real? Does reality take a mathematical form?

I will now provide a bit of my own way of looking at it.

For starters, we can say, by direct observation of our marbles, that we know that reality is not mathematically perfect. Reality is messy, not tidy. In reality, a marble released from the same place cannot even always end up in the same other place.

So science uses idealization and percent error. Thus a second answer to these questions is that we should not point to the success of science as a reason to believe that math is what is ultimately real. Science does not work that way. It does not use math in isolation from everything else, as if math alone is what makes thing happen. The arrangement of the setup (including tolerances) also contributes to a result.

But perhaps idealization is just one arcane example of how science works, and it is an exception to usual practices. So are there other specific examples of idealization in science?

Well, another example would be how moving objects are treated as dimensionless points in the equations of motion. Obviously, having zero dimensions is not real. But it is easier to do the math to find an answer that way, and then after we get the answer in our idealized situation of zero dimensions, we fill in the blanks of what is in the other dimensions as appropriate, whether it be a locomotive or a rabbit or a waterfall. Again, the math needs to be paired up with other aspects of reality (such as “baskets”). The math by itself is not the same as reality.

Other famous examples of idealization are the ideal gas laws, which come with their own formal method for how to correct for their giving you the “wrong” answers (they do not give you the answers that a measurement would give you). So you have to doctor them up with some specifics to the case in point.

Idealization is a genuine aspect of science, and it is not particularly controversial within science. In fact, it is usually recognized as a hugely empowering tool. But that is only until someone comes along and, as a matter of philosophy, insists that the idealizations are the same as ultimate reality, when in reality “reality” is all those (you know) wrong answers that deviate from the idealization.

So then that only leaves the issue of what to say to Plato. Couldn’t it still be argued (Plato would say) that, yes, the world is messy (it is not mathematically rigid), but the reason is that there is error in the world as it is trying to be perfectly mathematical? In other words, couldn’t it still be argued that the reason idealization works is that underlying it is how the world is actually based on math?

But I have already addressed that supposition in the example of scattering marbles on the floor. We could still deal with their random arrangement by drawing the best line through the marbles. Just because we can get a handle on a situation by applying the techniques of idealization does not in itself imply that the world is first of all arranged mathematically. It only says that we can do things such as set up coordinate schemes by which to measure anything.

And in any case, now with Plato we are conflating two different issues. One is to ask if the shape of all things must be mathematical (which I have answered “no” because idealizations are not the same as reality), and the second is to explain why the world is messy. As for this second issue (to explain the messiness) there are other answers than Plato’s when he holds that the world is messy because of error messing things up as they try to be perfect. And these other answers are more consistent with how science operates.

For instance, a common answer to why the world is messy is that the physical “laws” are only probabilistic in the first place. They are not mathematically rigid because what is fundamental is not math but how there are odds to things happening. And a second answer is that the “laws” often apply only to certain objects, not to everything, which makes things messy when we try to use them nonetheless in talking about “everything.” Newton’s law of gravitation, for instance, only applies to objects that are about the same size. And more generally, if we make a graph of the physical laws, that most usually creates a line that breaks down at the extreme ends of the graph, implying that the regularity only exists in certain places under certain conditions, not from being applicable everywhere (as would be the case if everything was premised solely on math).

These subjects (probability, and the domain-specificity of laws) are big topics so that I will not go deeply into them here. But I have already started to give a third explanation for why there is messiness in the world, so I will finish that now. Indeed, this third way can also start to explain the other two (probability and domain-specificity). It is this business of how science utilizes the math, not in isolation, but in the context of fitting stuff together to make things.

But wait a minute. Wait a minute. Do I mean that the real physical world is full of things like the baskets which keep things like marbles from spreading all over the place and so manage to create places which exhibit what looks to us like order and regularity in the world? And are the baskets (the places of arrangement) missing from the purely mathematical descriptions of reality?

Yes, that is exactly what I am saying. And that is feasible because the world is made of energy, and besides moving, energy also makes arrangements of itself (as in making “baskets”). It also explains why the order thus created is imperfect, since an arrangement of things can usually only produce imperfect regularity. (The second law of thermodynamics makes it so that arrangements must fall apart, so that they degrade and include irregularity). And some arrangements only make what is sometimes possible to happen (probabilities), and arrangements also make places within them where there is regularity and other places where there is not (domain-specificity).

So let’s look at an example of the “baskets” existing in nature.

It is possible, I suppose, to insist that the lungs are built on a plan of mathematical precision and that the air molecules, even as they are really moving randomly, are acting according to a math which is determining their actions. But it is more productive to see how the setup of the lungs—how they are organized—makes it so that the random actions accomplish the task of delivering the oxygen to the blood.

What is really happening is that the air molecules are moving randomly (not according to a law), and whenever objects are moving randomly, they tend to mix evenly. But because of how the lungs are organized, with a semipermeable barrier between the air and blood, that even mixing has the overall effect, in a slightly bigger picture than the random movements themselves, of transferring the oxygen from an area of high concentration (the lungs) to an area of lower concentration (the blood).

Order is born out of the random actions because of the arrangement of things in the lungs.

And then we can get mathematical about it, if we want to. We can measure concentrations and partial pressures and see how they are related. But the arrangement of the setup is what makes it happen. That is what is ultimately real. It is not that the math came first and made it happen.

For that matter, much of the world cannot be described by math in any case. That is because it is arranged in ways that are too complicated for math to describe.

Biology, for example, has only a few equations. And chemistry use many equations, but they are used to describe how countless random actions work together (not how each molecule is obeying a rule). And examples from physics are what this essay started with.

Evolution requires random actions in order to develop, but acting randomly is the opposite of following a mathematical prescription.

In human affairs, no one even pretends that issues such as whether Billy and Sue should get married is about mathematical formation.

It isn’t that the world takes on mathematical forms in the first place but that science—and nature—have found ways of dealing productively with the mess. (In past posts, I have discussed other examples of how the arrangement of things can transform random actions into order).

In a world where you can’t even roll a marble down an incline from the same spot and expect it to always land on the same different spot, this ability to act productively in spite of that is quite an achievement.

Isn’t it?

Science works by using math for making measurements. But measurement is a slightly different game from what the mathematicians play.

Measurement starts with defining the terms of measurement (which I haven’t discussed yet, but it can be another form of idealization since sometimes the terms are defined in a way that makes the math work). And how the terms are defined can be critically important because we end up understanding the world in the terms that we measure it by.

But coming up with these terms of measurement—as well as the idealizations, tolerances, and incorporating ways that things are fitting together—is different from doing pure mathematics.

We get into trouble whenever we conflate pure math with the activities of measurement.

Of course, once the math is established, it can be manipulated to derive new discoveries. And students learn to copy such derivations. But at some point, such derivations are founded on, and confirmed by, measurement.

With the theory that the math preexists existence itself, all it would take to conform a proposition would be to show that there is no error in the math (since all that counts for making reality would be the math playing out). That would mean we could prove that unicorns make rainbows just by making up an equation to that effect and having it contain no errors in the logic of the math. But the history of philosophy, for instance, shows us how ideas can be logically sound yet still very wrong.

So science measures.  If we want to call a proposition science, it cannot be about math alone without the measurement.