Aside from the lowest levels of physics itself, effectively everything that humanity has studied scientifically has been found to reduce to smaller physical parts. This idea is strongly backed by many, many, observations. Because this idea is strongly backed by evidence (which I will not list here), it follows that any idea which contradicts it has strong evidence against it.
This means that any theory which is compatible with reductionism and all our other observations ought to be considered much more likely to be true than those which are not. In this post, I will explain how to make sense of mathematics in a world where everything seems to reduce to smaller physical parts.
Reduction
In this section, I show that human mathematics can be reduced to physics.
Human brains, computers, and pen and paper are all reducible to smaller physical parts. Since all human mathematical reasoning is done using some combination of these things, it therefore follows that all of humanity’s experience with math has been reducible to smaller physical parts.
When we use math to predict something in the physical world, such as the trajectory of a moving object, we build a mathematical model that corresponds to the thing. The mathematical model, be it in the brain or on a computer, is reducible to smaller physical parts by our original premise. Since both ends of the correspondence are physical, the fact that they correspond to each other can be stated in terms of purely physical facts.
When doing math about things that might arguably not physically exist, such as infinity, the mathematical description of infinity encoded in our brain is still itself physical. By virtue of being a useful description, the description of infinity itself has properties which infinity would have if it existed. For example, the set of odd natural numbers would be infinite if it physically existed. We cannot make a bag of infinite apples. But we can make a program that returns true for any odd number (given arbitrary memory and time), and this physical (program) shares many properties with the infinite set of odd numbers mentioned in mathematics.
This approach works even for uncomputable things. We cannot make a bag of all the real numbers, but our mathematical descriptions of the real numbers share many properties that the real numbers would have, if they physically existed, and this is useful because many things in the real physical world work similarly to real numbers.
To be more clear about what I mean by mathematical descriptions “sharing properties” with the thing it describes, we can take as example the real numbers again. The real numbers have a property called the least upper bound property, which says that every nonempty collection of real numbers which is bounded above has a least upper bound. In mathematics, if I assume that a variable x is assigned to a nonempty set of real numbers which is bounded above, I can assume a variable y which points to its least upper bound. That I can do this is a very useful property that my description of the reals shares with the real numbers, but not with the rational numbers or the computable real numbers.
This approach is not really any different than how we talk about the number 3^^^3, even though there are less than 3^^^3 particles in the known universe. In both cases there are no physical objects corresponding directly to the description, but it still has useful or interesting properties. So if we accept that it’s reasonable to be able to talk about numbers larger than the number of particles in the known universe, then there is no need to reject any consistent mathematics.
Math Is Everywhere
With the observations from the previous section, we can see that all of human mathematics can be reduced to physics without loss. But it might still seem mysterious why mathematics seems to appear everywhere.
We will begin with noting that even though we as humans are embedded inside physics, we cannot learn about physics directly. There are only three ways for us to learn about physics: observations, inferences from observations, plus using a bit of knowledge built in to us by evolution. The built-in knowledge is inherently limited, so how we learn new things is almost always centered on *observations*.
The simplest form of observation is boolean, observing a process that returns either true or false. These observations can be combined using “AND” (∧), and negated using “NOT” (¬). Readers that know logic will observe that this is just propositional logic.
In fact, set theory, which can be used as a foundation for all of mathematics, assumes that we have some collection of discrete objects which we can make observations about, requires only one observation which is called “ELEMENT OF” (∈) which acts on two objects and returns true or false, the ability to do AND and NOT, as well as the ability to do these observations on every object at once using FORALL (∀). And that’s it.
That is *all* that is required to do *all of mathematics*. All of mathematics can be encoded as just taking some premises (observations) which use ELEMENT OF, AND, NOT, and FORALL, and figuring out which other observations are guaranteed if we have observed all the premises.
The point of noting this deep tie between mathematics and the concept of observation is that observation is effectively the only way for humans to know things about the world. This makes mathematics foundational to rational human thought. And a major part of the reason we find math everywhere is that we carry it with us everywhere.
No Mathematical Universe
The previous sections showed that mathematics can reduce to physics without loss, and provided an explanation for why mathematics shows up everywhere. This answered multiple questions that we might have previously tried to answer by positing the existence of a mathematical universe. But there are still questions left unanswered which might tempt us to believe in a mathematical universe.
For example, we might ask, why does the universe behave in an ordered way? However, I assert that a mathematical universe does not answer this question, only sweeps it under the rug. I say this because I could ask the same question about the mathematical universe: why is the mathematical universe ordered? We do not know the answer to that any better. Therefore nothing is explained.
I leave it as an exercise to the reader to apply this reasoning to any other question they might have which they think a mathematical universe might explain. In general I have not found anything which a mathematical universe explained which did not either contradict reductionism, or which could not be explained at least as well using the physical view of the world. I conclude from this that it’s unlikely that there exists a mathematical universe.
Extra: Math Is Underrated
I think I have done a decent job of being clear that I love mathematics in this post. Still, I want to say explicitly that I love mathematics. In fact I think math is underrated, even by most mathematicians. I think we should strive to know how to reduce all our reasoning to some foundation of mathematics, and that when we don’t know how we would do that reduction even in principle, that is a sign that we don’t understand something. This is a very high bar, but I think we need to aim high to have any hope of having properly justified beliefs.
Edit: My use of the word “reductionism” in this post has caused confusion, since I seem to use it in a non-standard way. The precise meaning I was aiming for is given in this paragraph of one of my comments here:
If we take as assumption that everything humans have observed has been made up of smaller physical parts (except possibly for the current elementary particles du jour, but that doesn’t matter for the sake of this argument) and that the macro state is entirely determined by the micro state (regardless of if it’s easy to compute for humans), there is a simple conclusion that follows logically from that.
Mathematics As Physics
Introduction
Aside from the lowest levels of physics itself, effectively everything that humanity has studied scientifically has been found to reduce to smaller physical parts. This idea is strongly backed by many, many, observations. Because this idea is strongly backed by evidence (which I will not list here), it follows that any idea which contradicts it has strong evidence against it.
This means that any theory which is compatible with reductionism and all our other observations ought to be considered much more likely to be true than those which are not. In this post, I will explain how to make sense of mathematics in a world where everything seems to reduce to smaller physical parts.
Reduction
In this section, I show that human mathematics can be reduced to physics.
Human brains, computers, and pen and paper are all reducible to smaller physical parts. Since all human mathematical reasoning is done using some combination of these things, it therefore follows that all of humanity’s experience with math has been reducible to smaller physical parts.
When we use math to predict something in the physical world, such as the trajectory of a moving object, we build a mathematical model that corresponds to the thing. The mathematical model, be it in the brain or on a computer, is reducible to smaller physical parts by our original premise. Since both ends of the correspondence are physical, the fact that they correspond to each other can be stated in terms of purely physical facts.
When doing math about things that might arguably not physically exist, such as infinity, the mathematical description of infinity encoded in our brain is still itself physical. By virtue of being a useful description, the description of infinity itself has properties which infinity would have if it existed. For example, the set of odd natural numbers would be infinite if it physically existed. We cannot make a bag of infinite apples. But we can make a program that returns true for any odd number (given arbitrary memory and time), and this physical (program) shares many properties with the infinite set of odd numbers mentioned in mathematics.
This approach works even for uncomputable things. We cannot make a bag of all the real numbers, but our mathematical descriptions of the real numbers share many properties that the real numbers would have, if they physically existed, and this is useful because many things in the real physical world work similarly to real numbers.
To be more clear about what I mean by mathematical descriptions “sharing properties” with the thing it describes, we can take as example the real numbers again. The real numbers have a property called the least upper bound property, which says that every nonempty collection of real numbers which is bounded above has a least upper bound. In mathematics, if I assume that a variable x is assigned to a nonempty set of real numbers which is bounded above, I can assume a variable y which points to its least upper bound. That I can do this is a very useful property that my description of the reals shares with the real numbers, but not with the rational numbers or the computable real numbers.
This approach is not really any different than how we talk about the number 3^^^3, even though there are less than 3^^^3 particles in the known universe. In both cases there are no physical objects corresponding directly to the description, but it still has useful or interesting properties. So if we accept that it’s reasonable to be able to talk about numbers larger than the number of particles in the known universe, then there is no need to reject any consistent mathematics.
Math Is Everywhere
With the observations from the previous section, we can see that all of human mathematics can be reduced to physics without loss. But it might still seem mysterious why mathematics seems to appear everywhere.
We will begin with noting that even though we as humans are embedded inside physics, we cannot learn about physics directly. There are only three ways for us to learn about physics: observations, inferences from observations, plus using a bit of knowledge built in to us by evolution. The built-in knowledge is inherently limited, so how we learn new things is almost always centered on *observations*.
The simplest form of observation is boolean, observing a process that returns either true or false. These observations can be combined using “AND” (∧), and negated using “NOT” (¬). Readers that know logic will observe that this is just propositional logic.
In fact, set theory, which can be used as a foundation for all of mathematics, assumes that we have some collection of discrete objects which we can make observations about, requires only one observation which is called “ELEMENT OF” (∈) which acts on two objects and returns true or false, the ability to do AND and NOT, as well as the ability to do these observations on every object at once using FORALL (∀). And that’s it.
That is *all* that is required to do *all of mathematics*. All of mathematics can be encoded as just taking some premises (observations) which use ELEMENT OF, AND, NOT, and FORALL, and figuring out which other observations are guaranteed if we have observed all the premises.
The point of noting this deep tie between mathematics and the concept of observation is that observation is effectively the only way for humans to know things about the world. This makes mathematics foundational to rational human thought. And a major part of the reason we find math everywhere is that we carry it with us everywhere.
No Mathematical Universe
The previous sections showed that mathematics can reduce to physics without loss, and provided an explanation for why mathematics shows up everywhere. This answered multiple questions that we might have previously tried to answer by positing the existence of a mathematical universe. But there are still questions left unanswered which might tempt us to believe in a mathematical universe.
For example, we might ask, why does the universe behave in an ordered way? However, I assert that a mathematical universe does not answer this question, only sweeps it under the rug. I say this because I could ask the same question about the mathematical universe: why is the mathematical universe ordered? We do not know the answer to that any better. Therefore nothing is explained.
I leave it as an exercise to the reader to apply this reasoning to any other question they might have which they think a mathematical universe might explain. In general I have not found anything which a mathematical universe explained which did not either contradict reductionism, or which could not be explained at least as well using the physical view of the world. I conclude from this that it’s unlikely that there exists a mathematical universe.
Extra: Math Is Underrated
I think I have done a decent job of being clear that I love mathematics in this post. Still, I want to say explicitly that I love mathematics. In fact I think math is underrated, even by most mathematicians. I think we should strive to know how to reduce all our reasoning to some foundation of mathematics, and that when we don’t know how we would do that reduction even in principle, that is a sign that we don’t understand something. This is a very high bar, but I think we need to aim high to have any hope of having properly justified beliefs.
Edit: My use of the word “reductionism” in this post has caused confusion, since I seem to use it in a non-standard way. The precise meaning I was aiming for is given in this paragraph of one of my comments here: