# What we can prove mathematically only applies to the system of logic used and mathematics, and not the world itself?

What we can prove mathematically only applies to the system of logic used and mathematics, and not the world itself? I am wondering if what we prove mathematically, only applies for the mathematical construct in which it was proved, which is abstract, but doesn't apply to reality itself, since we can prove that something that doesn't exist in the real world exists, but it doesn't apply to reality itself since in the reality it doesn't exist.

Meaning the reason why we think that what we prove mathematically must be true in the real world is because mathematics is an abstract construct and we think that abstraction allows us to generalize to say that it is also true of our world, but since we don't know what logic system the real world is based on, we cannot say with certainty that mathematical truths are truths within our world.

Is there any truth to what I am saying, or am I crazy?

• What we can prove mathematically applies to the world to the extent, and exactly to the extent, that the axioms we prove it from so apply. "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", Einstein. Sep 14, 2022 at 17:58
• Engineers successfully apply mathematics to the real world all the time. So do navigators, accountants, purchasers, sellers, and hosts of other people. Given this empirical evidence, your suggestion is obviously false. Sep 14, 2022 at 19:50
• The Kantian category of quality of your application of math theory to any real world problem could be said to be bridged and determined by the degree of reality defined as the degree of experience of a certain sensation in time, thus it hints this ultimately depends on your sensual experience... Sep 14, 2022 at 22:59
• Mathematical logic advocates claim logic is about form & validity. This is the topic's down fall. They do not say the content of the argument matters. How does mathematics prove unicorns do not exist formally? They don't . So all Mathematical logic does not refer to reality because math does not require content matter aka subject matter to exist. Mathematical logic allows empty sets. The real doesn't need to do so. Sometimes Mathematical logic applies to the real world & other times it doesn't. Take a valid argument with all false premise by content. How does that apply to the real world? Sep 15, 2022 at 11:52
• Possibly, what you are hinting at is some version of the formalist conception of mathematics, viz, that mathematical statements are not true or false until they are interpreted, and they are not true of the actual world unless they are given a real world interpretation. That is plausible, but the usefulness of mathematics within science and engineering shows that real world interpretations can indeed be found, at least approximately. Sep 15, 2022 at 19:27

It is not abstractness that causes us to think math applies to reality. It is utility. That is, we think it applies because applying it allows us to accomplish goals. And it allows us to predict what the outcome of situations will be.

However, it is not required for maths to be perfectly correct in order to apply to reality. This refers to the fact that measurement is always finitely accurate. As Feynman told us:

What is not surrounded by uncertainty cannot be the truth.

Consider the shape of the surface of the Earth. Our ancient ancestors had less ability to travel and so had less knowledge of the Earth's shape. A person who is only able to travel by walking is going to have a large challenge in this regard, particularly if those walks stay in the local valley. As transportation improved, knowledge of the Earth's shape improved. So by the ancient Greeks there was strong evidence the Earth was a ball.

Now consider the degree of accuracy involved. At one level of approximation, the Earth can be treated as a sphere. One can predict many things reasonably accurately. If your ability to measure distances was roughly that of the ancient Greeks, then a spherical shape would be difficult to distinguish from the Earth's actual shape from a sphere. In the age of exploration, it eventually became clear that the Earth's shape is a slightly squashed sphere. The equator bulges out a little due to the Earth's rotation. Some time later, with the advent of satellites, it became possible to measure the shape of the Earth to within a few meters, then within a fraction of a meter. Then to be able to detect sunami waves in mid-ocean where they are only a few cm high. And eventually, gravity wave detectors are able to detect stupendously small changes in the length of the sides of the detector.

But consider this progression.

At the first level, you can treat the surface of the Earth as flat. This allows such things as the arrangement of a farmer's fields. You make the sides "straight" and the angles 90 degrees, and you get rectangles. Straight in scare quotes because the sides are on the surface of the Earth which is not straight. But, to the degree of accuracy available to a farmer measuring a field a few 100 meters across, it is not possible to distinguish that non-flatness.

Later, the spherical model of the Earth is acceptable for a number of calculations. For example, predicting lunar eclipses is possible with a spherical Earth model. (You need to work out where the Earth's shadow will be in space.) The non-spherical nature of the Earth is small enough that it does not impact the typical lunar eclipse. Once again, it is a question of accuracy.

Later still, navigation required more accuracy. It was necessary to know that the Earth's equator bulges a bit, or you would get your path wrong when sailing across the ocean. Especially if you were channging your lattitude a lot.

Still later, in order to detect tsunami waves, we must be very much more accurate. But the passing of a gravity wave will not affect detecting a 20 cm high wave on the surface of the ocean.

At each stage, we apply maths to make predictions. Lay out farmer fields as though the ground were flat, and you can safely grow enough corn to feed your cows for the winter. Calculate your ocean voyage based on the slightly flattened sphere model, and you can safely sail from London to New York City. Model the surface of the ocean while neglecting gravity waves, and you can reliably provide warning of a tsunami on the way without producing false alarms.

At each step of accuracy, we can calculate things, we can make predictions, and those predictions are true. At that degree of accuracy.

The idea here is this. There is a shape to reality. Truth exists. We can only know it finitely accurateley. If we have an application that will function correctly within that degree of accuracy then we can apply mathematics to achieve our goals. The application of maths to such things is always done only up to some degree of accuracy. And that means we may not know the truth, but only some model of it that, within the accuracy we have currently, cannot be distinguished from the truth.

To return to the idea of abstraction. Each of these steps of accuracy represents a level of abstraction. The flat Earth model is an abstraction based on very short distance observations. The spherical model is an abstraction based on planet wide measurements of low accuracy. The squashed sphere a little more. The tsunami detector still more. Each level of abstraction has a range of applications which carry with them a range of accuracy. Choosing the degree of abstraction corresponds to choosing the range of application and the range of accuracy.

But it is the applications, not the abstraction, that makes us think mathematics applies to reality.