Saturday, January 11, 2014

Why “Nature Has Math” Leads to “Nature Has Software”

It has long been recognized by many scientists that in a fundamental way nature seems to have mathematics embedded within it. One widely discussed essay that commented on this was a paper entitled The Unreasonable Effectiveness of Mathematics in the Natural Sciences, written by the physicist Eugene Wigner in 1960. Wigner pointed out numerous cases where scientists took to using mathematical formulas to describe some little understood feature of nature, and such an approach ended up being successful beyond their wildest dreams. An example is Isaac Newton coming up with the formula for gravitation based on earthly experiences and a few astronomical observations, a formula which has been used ever since, and still holds up today as we study distant galaxies (notwithstanding the additional complications of general relativity). Another example is Maxwell coming up with some equations to describe electricity and magnetism, which also proved to be the right equations for radio waves, which were not even discovered until decades later.

Wigner stated, “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.”

This paper provoked widespread comment, and since then it has widely been stated that nature has math within it in some fundamental way. Let's use this as the starting point for a line of reasoning.

Premise 1: It is widely recognized by scientists that in some fundamental way, nature has mathematics kind of embedded within it, as shown by countless cases in which mathematical formulas are the correct way to describe fundamental regularities in nature.

There are numerous fundamental formulas of physics that could be cited to support this statement, such as Newton's law, Coulomb's law, the Schrodinger equation, Einstein's equation for time dilation, and Einstein's famous equation e= mc2. The nature of our universe is critically dependent upon such formulas.

Three formulas that dominate our universe

For an example of how impressed modern physicists are by the math within nature, you can look at this piece by physicist Max Tegmark entitled "Is the Universe Made of Math?"

Now let us consider: just what is mathematics? Mathematics is a very general term, which can refer to numbers, elements of geometry such as the circle and the circumference, and also simply logic, in the sense of mathematical algorithms and procedures. Most of what you learn in school when you study mathematics are procedures and algorithms for solving particular math problems.

In fact, mathematics seems to be mainly logic. Some famous philosophers such as Bertrand Russell and Alfred North Whitehead have argued that all mathematics can be reduced to logic, basically taking the position that math is just logic. A major branch of the philosophy of mathematics is called logicism, which maintains that all or most of math is just logic. So it therefore seems justified to state the next premise:

Premise 2: Very much of the math that we see operating within nature is a kind of logic.

If you doubt this premise, simply consider that almost every mathematical formula that appears in a physics text book can be stated in terms of logical statements such as “if/then” statements or algorithmic statements. For example, something such as Newton's law of gravity, which can be mathematically expressed by the formula shown above in the visual, can also be expressed by a logical statement such as “If two particles are separated by a particular distance, then the gravitational force of attraction between them is directly proportional to the product of their masses and inversely proportional to the square of the distance between them, and also proportional to a universal constant G.”

Now let us consider: what type of logic is it nature is applying as it behaves the way we observe it to behave? The answer is that nature uses a very precise logic of complete regularity. By way of contrast, consider the way logic is used by an entity much less regular, for example, a bigoted Southern sheriff of the 1960's. Such a person might have used logic in all kinds of irregular and unpredictable ways, perhaps applying the rule “If I see someone littering, I'll arrest him” if the person aroused the sheriff's prejudices, or if the sheriff was in a bad mood, but completely ignoring the rule if the person was someone who looked like the sheriff, or the sheriff was in a good mood. But nature doesn't act in that kind of mercurial or unpredictable way. When it comes to things such as gravitation, electromagnetism, and the Pauli Exclusion principle, nature applies logic in a completely regular, invariant, precise, and predictable way. The Apollo moon landing in 1969 depended critically on gravitation acting in a completely regular, invariant, precise, and predictable way. The mission designers knew there was absolutely zero chance that the astronauts would ever suffer from a “bad gravity” day caused by gravitation working in an irregular way.

Another aspect of the logic used by nature is that it uses constants, certain fundamental numbers that are always the same. Below are some of the most fundamental constants used by nature.

fundamental constants
Some fundamental constants of nature

So we must then state the next premise:

Premise 3: The logic we see operating within nature is a highly regular, invariant, precise, and predictable kind of logic, involving heavy use of fixed numerical constants.

Now we need merely ask ourselves: what are we talking about when we speak of the application of logic in a highly regular, invariant, precise, and predictable way, using numerical constants? This is not behavior that corresponds to the application of logic by human beings. Human beings are notoriously wavering and hard to predict, applying logic in all kinds of variant, biased, unpredictable, and inconsistent ways. The same political leader who will claim to be operating under fixed principles will then operate according to some other principles whenever it suits him politically.

But we do know of one type of system within our cities that does operate according to a highly regular, invariant, precise, and predictable kind of logic, making use of numerical constants. That system is a computer program. That leads to the next premise:

Premise 4: The application of logic in a highly regular, invariant, precise, and predictable kind of way, using fixed numerical constants, is a hallmark of programming.

This is the way computer programs operate. If I have a computer program that adds 1 to a total every time I press a key, the program will keep doing that with great regularity and predictability. It won't just do it 99 times out of 100. Computer programs also make use of fixed numerical constants, and all of the main programming languages have syntax support for declaring a constant within a program. For example, a computer program may have a line such as “const int AgeOfMajority = 18” which declares AgeOfMajority as a numerical constant. Then that constant will be used within the logic of the program, perhaps in a phrase beginning “if (AgeOfMajority >= 18).” 

It is a fair statement that nowadays when we see the application of logic in a highly regular, invariant, precise, and predictable kind of way, using fixed numerical constants, it is usually programming that we are seeing. 

Premise 5: It therefore seems that nature has programming.

This premise follows from premises 3 and 4, not as a matter of perfect certainty, but at least as a kind of likelihood (which is why I have used the word “seems”).

Now let's consider software. Is software something fundamentally different from programming? No. Most full-time software developers are called both computer programmers and also software developers. There's really no difference. So we can state this premise:

Premise 6: Software and programming are essentially the same thing.

Now from premise 5 and premise 6, this conclusion follows:

Conclusion: It therefore seems that nature has software.

This conclusion does not quite follow with metaphysical certainty, but this line of argument will do as a fairly simple bit of reasoning to support the claim that nature has built-in software.

For more elaborate arguments in support of the same conclusion that the universe must have built-in software, see my post The Theory of a Programmed Material Universe and my post Nature's Computation Needs Imply a Programmed Material Universe

As I explain in the first of these posts, once we realize that nature has built-in software, we can make much use of this idea as part of an explanatory framework, to help explain our universe's improbable evolution from an explosive beginning of supposedly infinite density to a place of fantastic harmony and order where life and Mind exist. 

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