# Why Are Physical Observations Mathematical?

Why does Newton's law of gravitation look the way it does? Why is the Gravitational Constant this specific value? Why do Maxwell's Equations look the way they do? Why is it that abstract quantities we make up such as mass, charge, displacement, time etc... could be put into useful formula that accurately predict phenomena in the physical world?

I am aware that physical laws are merely abstractions, but the mathematical correspondence between many physical quantities we define is deduced almost purely empirically i.e. is not abstractly defined by us as human observers. Let me emphasise that this is not necessarily the case with all laws, for example Newton's Second Law "F=ma," where we define the force to be the product of mass and acceleration, there is nothing about the law that is "empirically deduced." It is purely a definition. However, the Second Law of Thermodynamics, Faraday's Law and Special Relativity are just a few examples of where we deduce "mathematically meaningful correspondence," purely due to empirical observation.

Why is that so? Why does our universe seem to exhibit mathematically significant correspondence? Please keep in mind that I am referring to empirically deduced laws of physics, not quantities we define.

• Actually, your examples are very much abstract. One needs a very abstract notion of entropy to even formulate the second law, special relativity was a mathematically equivalent reformulation of Lorentz's ether theory, so it was by no means "empirically deduced". Indeed, nothing at all can be "empirically deduced", one needs induction or abduction to surmise empirical observations under a law or theory. So observations are not "mathematical". Why do physical observations lend themselves well to such surmises? Perhaps, physics covers just those kinds of observations that have that property. Commented Mar 13, 2020 at 23:43
• The asseriton that the non-conservative loop integral of the electric field is equivalent to the negative of the rate of change of magnetic flux is a purely emperical observation. The law is mathematically deduced from emperical observation. Very interesting view...physics covers the observations that could have inherent mathematical properties....this leads us to the question...can we construct a formalism that is not necessarliy physically meanginful that encompasses all inherent and non-inherent mathematically-significant phenomena i.e. a set of all possible physical theories? Commented Mar 14, 2020 at 0:29
• Observations know nothing about loop integrals, rates of change, electric fields or magnetic fluxes, those are mathematical and theoretical concepts. Only once they are in place and the equality is hypothesized can it be tested by observations, not "deduced" from them. No, we can not construct such a formalism as we have no clue ahead of time what sorts of phenomena might be out there, or what sorts of mathematics might be developed to surmise them. We do not even have such a thing for well-known complex phenomena studied in biology and sociology, which are at the bottom physical. Commented Mar 14, 2020 at 4:14
• @Conifold Yes but, theoretically, this would be possible, in outline, with abstract algebra and unlimited computations Commented Mar 14, 2020 at 5:31
• @CriglCragl I am not sure what "theoretically" means, it sounds like Kant's idea of constructing once and future science a priori, and did not go so well. If it means that God could do it then sure, but he has no need for it. Commented Mar 14, 2020 at 5:41

Let me emphasize that this is not necessarily the case with all laws, for example Newton's Second Law "F=ma," where we define the force to be the product of mass and acceleration, there is nothing about the law that is "empirically deduced." It is purely a definition.

This is not correct: there is a clear difference between a definition and a physics law, relating independently defined quantities. Mass and acceleration in the Newton's second law can be defined and measured independently from the law. Formal definitions of force may indeed appear as somewhat vague, but forces that are used in practice can be calculated independently - gravity force, electromagnetic forces, reaction force, elastic force, etc.

It is not the formal definitions, but the fact that the quantities are independently measurable, that matters to physicist: in this case a physics law translates into an equation, which allows calculating one quantities from knowing the other and making quantitative predictions about their behavior. In other words, physics is uses mathematics to make predictions on the basis of observations.

Note also, that, although forces remain an essential concept for engineering, modern physics usually operates with potentials and fields. Term force is then usually used not in its Newtonian sense, but as a synonym to interaction - see, e.g., Fundamental interactions.

It is not that physical observations are mathematical, it is that mathematics is a direct consequence of the way the universe is, that is, broadly, regular.

The regularity of nature was a condition for the development of neurobiological cognitive systems. The reason that neurological systems evolved is that cognition is a selective advantage. It is a selective advantage because nature is regular.

Natural selection produced a certain type of cognitive system. Essentially, our cognitive system is one which is based on deductive logic. Mathematics is only the formal extension of the human use of logic in everyday life. Our brain relies on logic to decide what we should do next. People survive because their cognitive system is logical, and also because it is efficient in terms of its energy requirements, response time etc. This, again, implies that nature is regular.

Thus, mathematics applies to nature because logical cells and neural systems evolved as an adaptative solution for survival. If our logic wasn't adapted to nature, we simply wouldn't exist to begin with. And once our logic is adapted to nature, our mathematics just follows from it.

Thus, the applicability of mathematics to nature, and to our observations of nature, follows from natural selection, which itself requires that nature be regular.

One implication is that mathematics is the product of a regular nature producing a logical capacity humans use to describe the regularity of nature. Nature describing its own regularity, so to speak. Or, regularities describing themselves.

Welcome.

Efficient abstractions are key to navigating a world we find ourselves in [1] , reducing cognitive load through predictive heuristics[2]. We developed our cognitive capacities for primarily social reasons [3], unlike smarter birds and cephalopods which seem to have been driven more directly by problem solving. Possibly for this reason we tend to interpret regularities as having an 'identity' we picture being 'out there' (because our cognitive machinery is based on theory-of-other's-minds), rather than just as patterns.

The most fundamental insight in physics so far, is that conservation laws, dimensions, and symmetries under translation, are directly equivalent [4]. Physical laws are patterns that provide predictive simplification, and similarities between sets of these imply similar patterns, and so simplifications, are present.

Blackholes appear to have maximum entropy within their envelope, they maximally resist simplification of their state, down to the Planck scale, every Planck 'bit' would have to be recorded for a full accounting (the holographic principle making that equivalent to a record of the event horizon). It seems like the universe is 'fundamentally' (ie, reconciling QM & GR) a spin-network of information exchange, with blackholes set at maximum entropy, Big Bang at minimum entropy and which can, at it's deepest level be interpreted in light of Wheeler's 'It From Bit' doctrine, as a series of 'yes' and 'no' answers, or regularities of a fundamental and unified kind. Hawking grappled with that being Godel Incompletel[5], but I would use Hofstadter's 'strange loop' model to infer that is because the universe has a fundamentally mind-like quality, where inquiry into it's own complexity increases it's own complexity, in a tangled hierarchy of self-reference.

Not to be too tongue-in-cheek about this, but you might as well be asking why a glove fits the human hand. I mean, walk into any department store anywhere in the world and buy a glove, and you'll get something that is basically hand-shaped. Weird!

The world is what the world is, and we write the best formulas we can to describe it. If we write formulas that don't fit the world — just as if we made gloves with seven fingers — we wouldn't be able to use them effectively, and so we'd eventually toss them in the garbage. If the universe were constructed a different way, we'd have developed different laws (maybe even a different math) that would fit that universe. There's nothing particularly mysterious about this.

Now, maybe it's a bit curious that the universe is consistent enough in the first place that we can write general laws about it. But even if it weren't consistent — if gravity or magnetism worked differently in different places, or changed behavior according to our mood — we'd still work out and formalize whatever consistencies we could, because the human mind is geared towards seeing, remembering, and analyzing patterns.

There is absolutely nothing unusual about a map being able to predict the terrain it maps. That is the purpose of a map; it is made to be that way. It only starts to look mysterious if we get confused and think the terrain somehow makes the map for us. The terrain doesn't need a map; we do. The universe doesn't need science; we do. See my point?

• The question is why a theory can be made in the first place. Under what formalism, can a physical theory be constructed and deemed valid, and why that formalism holds in the first place? Commented Mar 14, 2020 at 16:53
• @Joeseph123: I'm confused by this. All one needs to make a theory is an idea. I could sit here and theorize that the sky is blue because of countless tiny blue butterflies, and that's a perfectly fine theory: at least until it runs smack into empirical evidence. I think you're trying to put the cart before the horse — evaluate theories before testing them against evidence — and that's just going to tie you in philosophical knots. A theory's validity is not a matter of some preceding formalism; validity is a matter of successive results. Commented Mar 14, 2020 at 17:46
• The order of whether theory precedes or supercedes experimental verification is irrelevant to my question. Commented Mar 14, 2020 at 17:56
• @Joeseph123: Is it? How does that work? Or rather, what then do you mean by 'formalism'? There is a formalism (with a vast literature) about the proper method and practice of empirical science, but if you are not concerned about experimental verification, then what formalisms are you thinking about? Sorry, but I'm honestly perplexed... Commented Mar 14, 2020 at 18:08
• Let me rephrase...what "fundamental" principle of nature allows the scientific method to function, and even more importantly, how was this "principle" deduced to be a principle in the first place? Commented Mar 14, 2020 at 23:10

Here is my approach to this problem.

The "observations" that physics experimentalists make are for the most part not just observations but measurements of the system under study. The experimentalists are quantifying things like time, mass, length, charge, velocity and so on, and for them to have confidence in the reality of those measurements, they repeat them many, many times with the apparatus both in operation and idle.

This lets us reformulate the question thusly: why is it that the universe we inhabit is so constructed as to support consistent numerical measurements? Because it supports the consistency of the concept of number, and operations upon numbers like (at their simplest) addition and subtraction. In other words, there is no room for magic in our universe and by this I mean for example the ability to change the location of a material object at will without the performance of work, or to cause an object to burst into existence or vanish upon command, or to transport an object to a distant point in zero time. All our book-keeping has to balance out in the end; there is no free lunch; you can't take more kittens out of a box than were in it to begin with.

A universe that did not support measurement and arithmetic would be uninhabitable: if the distance to the sun from the earth in that universe were 93 million miles, then that distance would also equal a quarter of an inch or zero or a trillion miles a microsecond later or a million years later. If your box contained one kitten, then it also contained 43 kittens or a billion kittens or no kittens at all.