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Math Depends on the Physical Universe


  1. Introduction
  2. Apple Math
  3. Modern Math
  4. Super Apples
  5. Consensus and Probability
  6. Alternative Definitions of Mathematics
  7. The Axioms of Math
  8. Simple Ideas
  9. The Interdependence Between Mathematics and Physics

1 - Introduction

Many believe that mathematics and its theorems do not depend on the properties of our physical universe. They visualize mathematical systems as being completely independent of our universe, and valid in any hypothetical universe that may be imagined.

In this essay, I will explain how mathematics depends on certain properties of our physical universe, and then discuss some simple issues related to this dependence.

2 - Apple Math

The modern definitions of addition and natural numbers do not depend on the properties of apples. However, I wish to highlight some commonly overlooked properties of apples.

In this chapter, I define "apple math", which is a mathematical system that includes "apple addition" and "apple numbers". This system depends on apples. An apple number is defined to be the result of a process of counting apples in a box, starting with 0 when no apples can be seen in the box, and then 1, 2, 3, and so on. Addition, x + y -> z, is defined to be the process of starting with 2 boxes that have x and y apples in them each, then putting all those apples into a single box, and then counting the total in the final box.

Through experience, we learn that 2 + 1 -> 3. We've tried this many times over, and we always got the same result. The result is so consistent, that we no longer believe that it can ever happen any other way. We become so sure of 2 + 1 -> 3, that we consider it obvious.

Now let's talk about some of the physical properties that apple addition depends on. First it depends on our ability to count apples in boxes, and to combine them into boxes. Our ability in turn depends on our brain, and the physical properties of our brain, and consequently the physical laws governing the particles that make up our brain. Apple addition, also depends on the physical properties of the particles that make up apples. The laws of physics, are the laws that govern the particles that make up the apples and our brain.

Imagine, if you had 2 apples in a box, and then you add 1 apple into the box. A moment later, you count the apples to finish the process of apple addition, and the result is 4. "Preposterous!" one might exclaim. But it's only preposterous in our universe whose laws of physics happen to have certain properties, that make this process preposterous.

One of the properties that "apple addition" depends on is the stability of atoms, as well as the stability of the laws of physics. One can imagine a world, where the laws of physics would not be as stable as ours, and apples would be popping in and out of existence in our box. It is even easier to imagine, in our own world, a mathematical process called "bacteria addition", which would actually have similar properties to a hypothetical universe with apples popping in and out of existence. In the case of bacteria addition, our results will differ from regular apple addition, in such a way that they differ more greatly if you wait a longer period of time between the times you count the bacteria. Using "bacteria addition", we can define "fast bacteria addition" to be the result of some limit, which mathematicians reading this can easily devise. This makes it even harder, but not impossible, to imagine a hypothetical world with apples popping in and out of existence in such a way that no such limiting process is possible.

We don't even need the process of addition to explain the importance of stability for "apple math". We can look at 1 + 0 -> 0, or even 0 + 0 -> 0. For a good system of "apple math", we clearly need our apples to always have the same count, everytime we count the apples in the box. If we count 2 apples in the box, we expect 2 apples to be in the box at a later time. This property is also dependent on the stability of our universe. This is an often overlooked property of apple math, which is usually only noticed by children who are first introduced to apple math. Adults, on the other hand, take it for granted, and just assume that if they have a sandwich on their desk, look away for a second, and then look back, that the sandwich will still be there. Wait a moment, while a take a bite out of my... ah damn! You have to excuse me, I need to go make sandwich. I'm back now, with a new sandwich. Time certainly passes faster for those reading this essay, than for those who are writing it. In fact, the properties of time are also the properties of our physical universe on which apple math depends on very much.

Apple math depends on the properties of time. Our process of addition for example, takes place over the course of a finite amount of time. Apple math depends on time behaving in the way that we are used to, always moving forward in the way that we are familiar with. We would not be able to add apples, if, in the middle of our counting process, all the particles in our brain suddenly reversed their velocity, as though time in our brain was suddenly moving backwards.

We will not go into a complete analysis of all the properties of our physical universe that apple math depends on. We can certainly continue our analysis and find more properties. I think that for the purposes of this essay, the properties I've talked about so far, should be sufficient examples.

3 - Modern Math

It has been known centuries ago, that apple math depends on physical apples, and the properties of apples. Mathematicians have certainly taken efforts to remove the dependence of math on apples, and surveys seem to suggest that most mathematicians believe that they have also succeeded in making mathematics completely independent of the properties of our universe.

What does modern math look like today? Most of the math that I'm personally familiar with, takes the form of many large formal systems, each derived from a small set of axioms. For example, one book would define a vector space, and create a very large system by deriving dozens of theorems from those axioms. That is the most common form of math that I see. Some of the other things I am familiar with is how sometimes mathematicians will take one mathematical system, and derive theorems that are the same as the axioms of another system. In this way, they are able to unify formal systems into larger ones. Another thing I see happening, is mathematicians looking at a system, and creating another system which contains theorems (aka metatheorems) about the first system.

I claim that with all this stuff happening, the axiomatizations, the formal proofs, the metatheorems, and all that, math still depends on the properties of the physical universe.

4 - Super Apples

We can talk about two very important things, that make up apple math:

Instead of using apples, one can try to use something else. For example, pencil markings on paper. Most of the math today actually exists as markings on paper, including pencil markings in notebooks and printed markings in books. In second place, we have memories in human brains. In third place, we have data in computer storage. Computer storage will one day overtake the other two and be in first place, but as I'm writing this essay, it looks like it's still in third place, at least for non-numerical mathematics.

The machine that executes the algorithms of apple math is the human brain. In modern math, the human brain is also the main machine doing this. Computers can certainly perform the same algorithms, but as I'm writing this, it is still very unpopular. It seems that a lot of today's mathematicians are unable to trust computer software as well as they trust their own brains, or even the brains of other humans.

The modern rules of addition don't depend on apples at all. However, because mathematics uses markings on paper, memories in human brains, as well as the brain as a machine for executing algorithms, it still depend on properties of the universe. The markings on paper are just like a kind of "super apples". This comment also applies to the memories of mathematicians of whatever it is they visualize in their mind, whether they be the same symbols as they see on paper, or some more intuitive representations of the mathematical objects they use to make formal deductions with.

Mathematical symbols and intuitive representations of objects, are super apples.

When a mathematician derives theorems, he may start by writing down the axioms on paper. He will then execute algorithms in his brain, and then write down theorems he derives, one by one, on paper. When he is done filling up a whole page, he looks at the page, reads the results, and smiles as they correspond perfectly with the memories he still has in his brain, as well as his intuition. Just like in apple math, he takes it for granted that what he wrote might not change if he reads it again at a later time. He also takes it for granted that his memories will not change if he thinks about the results at a later time. Both his mind, and the paper, highly depend on the stability of our universe. The paper furthermore depends on the properties of the carbon atoms laid down on paper by his pencil. They also depend on the stability of the paper not to stretch, warp and morph around, which, in turn, depends on the properties of the 3 dimensional physical space. If the paper is attacked by a flying fireball, it will burn up, and all the results will be destroyed. If the paper is only partially burned, some of the markings may still be readable, but look different than the markings we will have in our memory.

The mathematician who runs to his friends to tell him about the new discovery he made, will certainly try to make sure that he isn't attacked by any flying fireballs. Mathematicians make the assumption that the markings on paper and the memories in their brain are stable, without realizing that this assumption is derived from the physical properties of the universe. Just as it is easy to imagine a universe with apples popping in and out of existence, it is possible to imagine a universe that doesn't have any kind of paper stable enough to derive the same mathematical results that we have derived in this universe. Mathematics, as we have developed it, depends on certain properties of this universe. Not all the properties of the universe, but at least some.

5 - Consensus and Probability

I can imagine a mathematician of the past looking at the properties of apple math, and seeing that they have the same properties as orange math and banana math. Instead of having many different kinds of math, that seem to agree with each other, he can define a standard based on apples, and say that "standard addition" is based on what happens in the processes of "apple math". He can then create physical theories that claim that the results of "orange math" are the same as the results of "standard math". He can then use "standard math" to derive theorems and results, and then make predictions about what would happen when we put oranges into boxes and try to count them. These predictions are about the physical world, so are clearly physical theories that depend on the properties of the universe. It is just like how physicists and other scientists use math to make predictions about the physical world. The "standard math" in this case, is derived from apples, so it also depends on the properties of the universe. Just because any fruit can be used to derive the same theorems of addition that "standard math" has, it does not mean that "standard math" is independent of fruit, it is still dependent on fruit, however, it is only dependent on one particular fruit, namely apples. So even though it is still dependent on fruit, it is much less dependent on fruit that it was before.

In the modern case, we have, Mr Apple and Mr Orange. Mr Apple is a mathematician, and he derives certain theorems based on a set of axioms. Mr Orange, takes the same axioms, and tries to do the same derivations as Mr Apple has. They do these derivations independently, but later meet up and compare their results. "Success!" Mr Apple shouts, "We have derived the same results, therefore I believe these results do not depend on either myself, nor on Mr Orange. These results are completely independent of me, Mr Orange, and everybody else!" Then along comes Mr Turd! Mr Turd proves him wrong and ruins Mr Apples excitement. Mr Turd takes the axioms, tries as hard as he can to make the derivation, but keeps getting a different answer. Mr Apple tries to explain to Mr Turd what he's doing wrong. But to no avail, Mr Turd's brain just keeps coming up with a different result.

Mr Apple decides to make a proclamation. He states that the results that he, Mr Orange, and other mathematicians who come up with the same results are to be considered correct, but that the results of Mr Turd, and others who smell like him are to be considered incorrect, and not called mathematics. Has Mr Apple succeeded in making mathematics independent of himself, Mr Orange, and everybody else? Not at all. All he has done, is define a standard. This is no different than the mathematician of the past who has defined "standard math" as being the results derived from "apple math" rather than any of the other fruits. So math no longer depends on Mr Turd, but it still depends on the others who form a consensus on which results are to be accepted.

Standard math of today, relies highly on consensus, as well as on each mathematician's personal beliefs. Not all mathematicians have verified the proof of Fermat's last theorem, yet many of them have made a decision on whether or not to believe that theorem, which is at least partly based on the consensus of other mathematicians that have verified the proof themselves.

Our universe must have certain properties, such as more than one human alive, in order for consensus to even work. Furthermore, if our universe follows the laws of probability very well, then with enough mathematicians to verify the proofs of theorems, consensus works incredibly well. Mathematicians believe the results derived through consensus and repeated derivations of the same results, because they believe in certain laws of probability, as well as on the ability of mathematicians' brains to work correctly most of the time. All these things also depend on the properties of the universe. Most mathematicians also believe that if they attempt the derivation of a theorem again in the future, that they will get the same results, just as they believe the sun will rise the next morning. However, the manifestation of the beliefs depends on a property of stability of our universe, the properties of time in our universe, and more.

It should be clear at this point that relying on consensus, or the repeated derivation of the same result, does not make mathematics independent of our universe. Consensus certainly makes it less dependent on individual human brains, and individual derivations, but not completely independent of our universe.

6 - Alternative Definitions of Mathematics

The definition of "mathematics" I have used to far is the common definition of mathematics. That is, mathematics, at least, includes the writings in math books, includes the things people do when we say they are "doing math", and includes the things people look at when they say they are "studying math".

There exist various alternative definitions of mathematics, which are in contradiction with the common definition. Is it possible that some of these alternative kinds of mathematics do not depend on the properties of the physical universe?

Formal logic, just like mathematics, also depends on the properties of the physical universe. Any logical, or intuitive thought process we have also depends on our universe. Therefore, any definitions we come up with, also depend on the physical universe. Our definitions of any word depend on stability for the definition to remain the same over time. So any alternative definition of the word "mathematics", depends on the properties of our universe. Therefore, in search of a kind of mathematics that does not depend on our universe, we need to examine, not the definition itself, but the thing the definition is trying to point at, that is, the thing it wants us to look at and see, or imagine, when we use the word "mathematics".

I have seen people try to define math by pointing to something that is beyond our universe. Some of them usually mention something about the angles of triangles always adding up to 180 degrees, regardless of which universe we would be in. Physical triangles, clearly could not exist in a universe that is one-dimensional. Axiomatic triangles, may exist in a one-dimensional universe, represented by symbols, but they still require some kind of property of stabilitity, to derive theorems about the triangles. Clearly, even this simple theorem about triangles would require certain properties similar to the properties of our universe. So it does in fact depend on which universe we would be in. If we would happen to be in a universe that has apples popping in and out of existence, we might not be able to derive the theorem that the angles of triangles add up to 180 degrees. A related problem is that intelligent life may not even exist in a universe as unstable as that.

Instead of analyzing the attempts of others to define mathematics in a certain way, and checking whether or not they depend on our universe, let's try to simply define mathematics to be independent of our universe. Here is one attempt: "Mathematics is the collection of theorems that do not depend on the physical properties of the universe". This definition, however, based on a certain interpretation of the word "theorem", may lead to a contradiction. Theorems, as we know them, depend on being "true" or "always true". If time exists, the theorems must be stable over the passage of time. Therefore, if time exists, then theorems depend on the properties of the universe, contradicting the definition itself. Therefore, to avoid a contradiction, one of the things that must be assumed about that kind of mathematics, is that it doesn't even depend on the existence of time. This kind of mathematics must exist in a place that is "beyond time", if that term can even be interpreted in a sensical way. I find it hard to imagine that this kind of mathematics, or the kind of theorems it contains, can exist. This is really pushing math towards the realm of the supernatural, not unlike the supernatural ghosts imagined by frightened children. I have yet to see a concrete definition of this kind of "ghost math" that is unambiguously communicable from the creator of the definition to another person with a sound mind.

Let us now abandon, for the rest of this essay, any attempt to define math in such a way that it is independent of the physical properties of the universe. Let us instead, say that mathematicians should strive to make mathematics as least dependent on the properties of this universe as possible.

7 - The Axioms of Math

One possible attempt to object to my claim that math depends on the universe is the following argument:

  1. Mathematics is completely derived from a set of axioms.
  2. The axioms do not refer to anything in our universe.
  3. Therefore, mathematics is independent of our universe.

Let's assume for a moment that the first two statements are true. Even if the first two statements are correct, the third step does not follow from them. The third step would have to be "Therefore, the theorems do not refer to anything in our universe." The axioms and theorems might not claim to refer to anything in the universe. However, the statement that mathematics depends on the universe is not part of the formal system in which the axioms and theorems take place. When we talk about mathematics depending on the universe, we are stating a metatheorem in a metatheory that is placed at certain level above the axioms and theorems of math. We should not attempt to use the axioms and theorems of math to talk about math itself in a circular way. To talk about math, we need a metatheory that lies outside of the theories of math, at least one level above it.

There are problems with the first two statements as well. In the second statement, the axioms might not state that they refer to anything in our universe. An axiom may say: "A group is a set with a binary operation such that..." and not "A symbol representing a group is a set with a binary operation such that..." However, even though the axioms don't utter the word "apple" or "symbol" or "markings on paper", they do in fact refer to certain symbols that may later be drawn, as well as the intuitive objects mathematicians will have in their mind that correlate to those symbols.

As for the first statement, it's clearly a redefinition of the word mathematics as it has been used for centuries. Mathematics has not always been as formal as it is today, and if you decide to redefine math to exclude all non-formal methods, then you are creating a personal definition that is in contradiction with the common definition.

Any attempt to remedy the example argument I have written above would certainly not remedy the first point I made, that a statement about the theorems of math would be a metatheorem, and not one of the theorems themselves.

8 - Simple Ideas

A simple, but possibly important, idea that can be seen when we acknowledge that mathematics depends on the physical properties of the universe is the idea that mathematicians should strive to make mathematics as least dependent on the properties of this universe as possible, or, to make it depend on the least amount of such properties as possible.

Mathematics depends on taking theorems and executing algorithms over the course of time to derive other theorems. So certainly, we need to depend on the existence of time, and on the stability of our theorems and algorithms over time. However, the current dependence on the human brain, and all the particles making up human brains is something that we can certainly improve on. The objects of our universe as far as we can currently see depend on a very small set of laws of physics. However, the brain, which follows those laws of physics is made up of a very large number of particles, arranged in a very complicated way, with presently unknown boundary conditions, making it very difficult to model and study. We should therefore, put more effort on having computers execute all of the same algorithms that the brains of mathematicians currently do. Doing so, will not only make math less dependent on the complicated human brains, but will also facilitate the study of mathematics itself.

Understanding that mathematics depends on certain, but not all, properties of the physical universe, also leads to the simple idea of studying those properties. What are the properties in question? How many can we find? Is this set of properties complete? Can we modify math to decrease the amount of properties that it depends on? Which properties should math depend on? Which properties should math not depend on?

When developing computers with the goal of replacing human brains for executing the algorithms of math, we may study the physical laws governing the central processing unit in the computer. We may try to design the computer in such a way that it depends on as few laws of physics as possible.

A question for which there is no clear consensus, is whether or not mathematics is a science. I hope that in view of my essay's explanation of how mathematics depends on certain properties of the physical universe, that people will switch sides, and consider mathematics to be a science, just like I always have.

9 - The Interdependence Between Mathematics and Physics

Every physicist and mathematician acknowledges that physics depends on some mathematical theorems. However, we now see, that mathematics also depends on some laws of physics.

Mathematics and physics have always depended on each other. History is full of examples of people developing new mathematical ideas while studying physics, and people developing new physical ideas while studying mathematics. Even when apple math was first invented by Mr Apple, he developed a law of physics about apples remaining at rest, and not floating away.

I believe it is important for scientists to see and study this interdependence.

Scientists should study the interdependence between mathematics and physics.

Math Depends on the Physical Universe, by The Humanoid