In the last episode, I discussed one way in which words gain their meaning: definition by reference. When a community agreed to use the same word to refer to the entities with some shared similarities, the word became the reference to those entities. Words like “cat”, “dog”, “cloud”, and “rainbow” are some of the examples of words defined by reference. But definition by reference can only apply to things that we can use our finger to point to. Many words simply can not be defined in this way. For example, how can we use references to define the word “father”? We can point to pictures of men, but how can the learner know that we meant “father” but not just “men”? One way to clarify this is to use pictures with an older adult male and a younger child and point to the older male in the pictures. Let’s leave aside the question that whether this method is sufficient to teach someone the concept of “father”. But this does illustrate something interesting. Even though, we only point to the older adult male as the reference to the word “father”, the presence of the younger child is crucial in defining the concept of “father”. Without the younger child, with only the older adult male by himself, we can not differentiate the concept of “father” from the concept of “man”. The concept of father does not refer to specific people or entity, but a member of a group in which different members have specific relationships with the other members. In this case, a father is a member of a family group that is bonded by a father-child relationship. The child is the other member of the family group. The existence of a father predicates the existence of a child. Similarly, the existence of a child predicates the existence of a father. That is when we are talking about a father, we are implying the existence of a child to that father, even though we do not explicitly talk about that child. We, humans, are constantly making those hidden assumptions when using words that are defined by relationships both consciously and unconsciously. When we use the word poor we imply there is a state of abundance that we hope for. When we use the word injustice, we imply that there is a potential state of fairness and equality. This “baggage” human language carries, makes our communication not only dependent on what is said but also what is not said in context and subtext.
So, different from the concepts that are defined by reference, concepts that are defined by relationships are only valid within the group that it is defined in. For example, the concept of cat is defined by reference. Therefore, no matter in what context I am speaking, I can always refer to the furry animal with pointy ears as a cat. But It is not true for concepts that are defined by relationships. This was very well reflected in one of Plato’s dialogues, Euthydemus, which contrasted the difference between rhetoric and philosophy, Sophism and Socratic dialogue. In the dialogue, The Sophist named Dionysodorus asked if Ctesippus has a dog, and Ctesippus answered, “yes, a real rogue”. Then Dionysodorus asked: “Has he got puppies?”. “Yes”, answered Ctesippus, “they are rogues just like him”. “So the dog is a father then?” asked Dionysodorus. “Yes”, answered Ctesippus. “And the dog is yours?” asked Dionysodorus. “Certainly”, Ctesippus answered. “Thus he is a father, and yours, and accordingly the dog turns out to be your father, and you a brother of whelps.”, concluded Dionysodorus.
Philosophers often get a bad reputation for making ridiculous statements. There is another famous statement called the Dichotomy paradox proposed by Zeno of Elea, you probably heard it before. Suppose you want to cross a street. To get to the other side, you have to first walk halfway across the street but to get to halfway across the street, you have to first walk a quarter way across, but to walk a quarter way across, you have to walk 1/8 of the street and so on. In this way, we can divide walking across the street into infinitely many small tasks, and each takes a finite amount of time, but adding infinitely many of the small tasks together would be an infinite amount of time to cross the street, therefore, you can never reach the end of the street. The story was told that Diogenes the Cynic, once heard of the paradox, stood up, and walked across the street to demonstrate the falsity of Zeno’s conclusion. But Diogenes has missed the point. Zeno certainly knew that he can cross the street before the end of the universe, but what he was curious about, was which step of reasoning he was mistaken that has lead to the false conclusion. That turns out to be a far more interesting question. It was after 2000 years, Newton has demonstrated the sum of some infinite series of numbers can be finite using calculus. I said demonstrated, not proved, because even though Newton’s calculus does the correct computation, a lot of Newton’s statements regarding its validity was based on self-evident arguments. It was another century until Bernard Bolzano and Karl Weierstrass provided the formal proof.
When we examine a statement, we should not only check if the conclusion is right or wrong but also get curious about why. If we think the conclusion is right, we should be able to provide logical arguments and evidence to support it. If we think the conclusion is wrong, we should be able to point out to our interlocutors where they have made the logical error. To examine that we have to think from their perspective and reason based on their beliefs and premises. As Bertrand Russell has pointed out, if someone’s conclusion is incompatible with our own beliefs, we can only say that subjectively, we do not like it. Only when we have found logical contradictions within someone’s belief system, we may objectively conclude, that they are wrong. This process demands us to try to understand from other’s perspectives, it also makes us examine our own arguments to avoid making the same logical errors in our reasoning. In Plato’s dialogue, Euthydemus, it was obvious that Ctesippus’s dog was not his father, but what is far more interesting, is where the error of logic occurred. This comes back to how the concept of “father” is defined. Ctesippus’s dog is Ctesippus’s. And the dog is a father, but only in relation to its puppies. Unlike the word cat, that a cat in any context is a cat, an entity in some context is a father, but in others is not. When we talking about the dog’s relationship with the puppies, it is the father. But when we are talking about the dog’s relationship with Ctesippus, then the dog is not the father. This is the mistake Sophist Dionysodorus has made.
Languages carry baggage. There are the implicit meanings we may or may not intend to express when we speak sentences. For example, when I say, “This is the father.” I imply there exists at least one entity that is the child of the “father” whom I was referring to in the sentence. Here is another example, if I say, “I got a new computer”, most people will interpret that I had an old computer which I am replacing with the new one, even though I did not explicitly say I had an old computer in my sentence. In the two examples above, it may seem obvious what baggage those sentences carry. But in a more abstract context especially with words without a good definition, it can become hard to tell what is the baggage that the sentence is carrying. For example, when I say “This is a good choice”. What does the word “good” implies? Does it mean that the choice is aligned with a standard that we consider as “good”? If so, what makes the standard “good”? And does it exist a bad standard that is in opposition to “good”? How can we define good? I know that for many people if they were asked about what is good, they probably will use some words that are synonyms to good to define good such as “desirable”, or “right”, but those definitions only push the problem down more, what does “desirable” or “right” mean? I will not dive too deep into the philosophy of ethics here, as I am discussing the philosophy of language, but I hope I have demonstrated that when we use a word as simple as “good”, we might be unconsciously implying a lot of contexts without even realizing it. I do want to invite you to think about how are abstract concepts such as “good” are defined: are they defined by references or relationships? Or something else? Can such concepts pass the linguistic agreement tests? Do humans agree on what those concepts mean? If so how can we know that we agree?
Now back to the philosophy of language. I have introduced the idea of definition by relationships, allow me to make the attempt to define numbers here. Numbers are quite fascinating, in countries with compulsory educations, almost all members of the society know how to add or multiply numbers. If you think carefully, you will realize that for most of us, we only learned in school how to do operations with numbers, but never been taught the definitions of numbers. Some may say that numbers are 1, 2, 3, 4, etc. But those are examples of numbers, not the definition of numbers. Similarly, if you ask me what is a chair, by pointing to a chair, I have only given you an example of a chair, not what a chair is. A definition for a concept may not always exist, as we have discussed in the last episode, there are many concepts we simply do not have a logically consistent definition and as Wittgenstein has pointed out, the meaning of words changes constantly when used differently by different groups. But when a definition for a concept does exist, it should be able to tell us, unambiguously, if an entity we examine fits the definition, therefore is an example of the concept, or not. Given that it seems so obvious that we can unambiguously tell if something is a number or not, many philosophers believed that definition for numbers does exist and made many attempts to find it.
In ancient Greece, when algebra has not yet been invented and all mathematical proofs are done geometrically instead of algebraically, Plato thinks of geometric shapes, such as perfect squares, perfect circles, and perfect triangles, as real objects in an ideal world and all imperfect squares, circles, and triangles in our world are mere shadows of the shapes in the higher reality. This is quite similar to how Plato understands other concepts. He also believes the perfect version of a knife or a cat, existing in a higher reality, and when we see a knife or cat, we are comparing what we see to the perfect version and determines if what we see is a knife or a cat. This kind of makes some sense for things that we can see and touch, but it raises another question: what is the Platonic ideal of the number one in the higher reality when we can use it both for one cup of water and one meter? What is the commonality shared between one cup of water and one meter?
Philosophers such as John Stuart Mill argued that numbers are the property of things, like color or hardness. For example, if I see 3 apples, the 3 is the observed factual property of the group of apples. Bertrand Russell countered this definition by giving the following example: taking a pair of shoes, by how we are counting them, if we count individual shoes, then there are two of them, but if we count pairs of shoes, there is only one pair. Yes, at different times and different lighting conditions, the cloud can show different colors. But under the same condition, objects should always display the same color. But for a pair of shoes, under the same condition, we can think of it as one or two simply based on how we are counting them.
Gottlob Frege addressed this issue by introducing the theory of types. Frege defines numbers as extensions of concepts. He said: ‘The number of F’s’ is defined as the extension of the concept G, such that G is a concept that is equinumerous to F‘. But what does that even mean? Indeed, Frege used words such as “extension”, “concept “, and “equinumerous“, and we will have to know what they mean before we can understand his definition of number.
This leads to a very interesting problem. In “Introduction to Mathematical Philosophy”, Bertrand Russell has pointed out, correctly, that if all words are defined by other words as the dictionaries do, then we unavoidably will end up either with infinite regress or cyclic definitions. So we have to be able to give some words or concepts meaning without using other words. We can only achieve this, through definition by reference. Consider the definition of a cat, how can we define the concept of a cat to someone who can not see, hear, or touch? We can describe how it looks, but they won’t be able to understand it because they can’t perceive sight. We can describe how it sounds, but they also won’t understand it as they can’t perceive sounds. It is not just the words that describe the sound of a cat that won’t make sense to them, any words about any sound won’t make sense to someone who can not hear. There is no way we can communicate the idea of a cat based on our perception of the cat to this person. What is a cat to us is just the sum of our sensory experiences when we interact with a cat. How it looks, sounds, smells, feels. When we think our sensory experiences are close enough to the reference we used when we first learned the concept of a cat, we would conclude that it is a cat that we are interacting with.
In psychology, our subjective experience when perceiving a physicals phenomenon is called Qualia. When we use the definition by reference for words, we are putting a label, in this case, the word “cat”, to a collection of certain sensations, or Qualia that we have when interacting with those references, or in this case, cats. When defined in this way, the concept of cats is not defined by any words, but by my sensations, I know a cat when I feel that it is a cat. Many philosophers have argued that we can never use words to communicate Qualia, it has to be understood through the first-person experience. For someone who is blind, no words can be uses to make them understand the concept of a rainbow. Certain knowledge has to be gained through interacting with the physical world using our subjective senses not from reading books or learning from others, and those concepts can only be defined through a reference.
This is what I think the fundamental limit of machine learning. It is not that computers are not as capable of doing computation as humans. For that, we have many many examples of the superiority of the computer’s power to reason over humans. For instance, the computer Deep Blue defeated world champions in chess in 1996, and modern computers have become literally hundreds of thousands of times faster in comparison. What is the fundamental limitation of machine learning is that machines do not share the same sensory experience as humans, therefore they lack the Qualia to understand the world as we do. To think how difficult it is, for a person in a privileged position to empathize with someone who is in poverty, and those are two humans who share the same sensory organs but just have different experiences. How can a machine, that does not sense the world as we do, or does not feel as we do can see the world the way we see it? This, of course, does not mean machines can not be intelligent or understand the world, it just means that they will not understand the world in the same way as we do.
Now, let’s get back to the definition of numbers. Allow me first to give 6 concepts without providing any word definition. I will demonstrate later how we can use references to put those concepts into context and give them a concrete meaning. We call concepts without word definition, primitive concepts. The 6 primitive concepts of arithmetic are “1”, “the set of numbers”, “equals to”, “is in”, “if…then…”, “add”. Those concepts are defined by the following relationships:
- 1 is in the set of numbers.
- If x is in the set of numbers, then x add 1 is in the set of numbers
- If A add 1 equals B add 1, then A equals B.
These three relations should be fairly simple to accept by anyone who is familiar with arithmetics. In plain English, the first rule just says 1 is a number, the 2nd rule says if x is a number x+1 is also a number, and the third rule says that if A + 1 equals to B + 1, then we know A = B. Note those rules above only declare the relationship between the 6 concepts: “1”, “the set of numbers”, “equals to”, “is in”, “if…then…”, and “add” and they are composed only of those 6 concepts. But those relationships do not tell us what those concepts mean in our physical world. But I will get there. With those relationships defined we can extend our definitions to more concepts, for instance: 2 is 1 add 1. Then we can define 3 as 2 add 1, and so on for 4, and 5, and all the other numbers.
Let’s say we have 1 orange. We can say it is a pile of 1 orange, though it is not much of a pile. So we toss in another orange. Now it is a pile of 2 oranges. We have learned in elementary school that we can use arithmetic to calculate the number of oranges in a pile. For example, if I have a pile of 4 oranges, and a pile of 5 oranges, when we combine them, without counting them, we know we will get a pile of 9 oranges. But here is an intriguing question, how can we know for sure there are 9 oranges in the combined pile without counting them, but only from a calculation? This is the central question surrounding the relationship between mathematics and the physical world. How is it possible that we can know something about the real world only based on my prior knowledge and some seemingly made-up mathematical rules, without actually examining the real world? How can we get the number of oranges of the combined piles, without actually counting it, just from our knowledge of the two original piles and the rules of arithmetics? Furthermore, if we never counted the pile of combined oranges, how can we be sure that there are 9 oranges in it? Why should it be 9 oranges in the piles? The oranges don’t crunch the numbers when we combining them together and decided to be 9 so they will obey the laws of arithmetics. Oranges don’t know about the law of arithmetics, they just stack together. It turns out to be a rather complex question, that why arithmetics can be used to model and predict the number of orange in a pile.
We learned in school that we can use arithmetic to get the number of oranges of the combined pile because it is self-evident or just intuition. I would agree that the result is 9 oranges is intuitive to me as well, but that does not answer why 4 + 5 = 9 would give us the correct number of oranges without counting the combined pile explicitly. Furthermore, I have watched kids learned arithmetic, and it was neither intuitive nor simple for them when first exposed to those concepts. Only after hundreds and thousands of practice problems, they start to gaining intuition about arithmetic, this is also true for other knowledge. So intuition is much based on how familiar we are to a concept, and once we are familiar with it, it is hard for us to remember what it feels like to be foreign to those concepts when we first started learning them. Furthermore, for many people, general relativity or quantum mechanics are absurdly bizarre and completely counter-intuitive, but those mathematical equations can provide extremely accurate predictions about the physical world.
The key to understand the relationship between mathematics and the physical world is how mathematical concepts are defined. As given above, arithmetic can be defined by 6 primitive concepts: “1”, “the set of numbers”, “equal to”, “is in”, “if…then…”, “add”; and three relationships:
- 1 is in the set of numbers.
- If x is in the set of numbers, then x add 1 is in the set of numbers
- If A add 1 equals B add 1, then A equals B.
Those relationships tell us nothing about what those concepts are in the physical world, because, in a different context, they mean different things. Take our piles of orange for example.
For relationship 1: We use the symbol 1 to refer to a single orange. The concept of “the set of numbers” refers to all the possible piles of oranges. The concept of “is in”, as “A is in B”, means “A is one of the possibilities of B”. Then the first relationship: 1 is in the set of numbers, in this context, means that a single orange is considered as a pile of oranges out of all of the possible piles of oranges.
For relationship 2: We use the concept of “if A then B” to indicate a logical connection between two situations, that is when we observed situation A, then we know B must be true even if we have not directly observed it. So the relationship: If x is in the set of numbers, then x add 1 is in the set of numbers, it means that when we have a pile of oranges, after putting in another orange, it is still a pile of oranges. The concept of “add” here refers to the action of putting additional oranges into an existing pile of oranges or combining two orange piles together.
For relationship 3: We have two piles of oranges, we put one additional orange to each pile. If after the action, two piles have the equal amount of oranges, even though we have not counted before adding the additional oranges, we can still conclude that the two piles have the equal amount of oranges before the additional ones put into the piles. One way to define “equal” is to use the following action: if we take one orange from pile A then pair it with one orange from pile B, put them away, repeat the action, until either pile A or pile B has 1 orange left. If both of them have 1 orange left, pile A and pile B are equal. If one of them has 1 orange left, and the other one does not, they are not equal. I used this pairing action as the reference to “equal” here instead of counting because counting oranges requires the use of general numbers, such as 2, 3, 4, etc., which are not part of the 6 concepts and 3 relationships given above. So we have not yet defined general numbers. Therefore, we can’t use them to define the concept of “equal”.
In the example above, I have contextualized the 6 primitive concepts of arithmetic in the scenario of orange piles. In different situations, whether counting cups of water or the number of shoes, those primitive concepts will be contextualized by different physical references. It might be bluntly obvious how to count oranges, and it may seem that I am making things overly complicated for something as simple as 2 + 2 = 4. But I will show you why it is important to define mathematics as a set of primitive concepts and relationships that is detached from the physical world and those primitive concepts and relationships should only be contextualized in each application separately. There is an idea called indexical in the philosophy of language. An indexical word is a word that refers to different things in different contexts. For example, the word “here”. “Here” for me, at the moment is the apartment I am living in. But “here” for you, is where you are at the moment. Similarly, numbers are indexical. There is no inherent 2ness of the shoes when we say there are 2 shoes on the floor. We say there are 2 shoes on the floor because we have chosen implicitly to use a single shoe as the reference when we count shoes in this context. If we have chosen to use a pair of shoes as our reference of 1. Then there is only 1 pair of shoes. The meaning of 1 and 2 changes based on what context we choose, or in this specific case, what unit we choose.
Let’s examine the question: why is 2 + 2 = 4? In elementary school, we learned that 2 + 2 equals 4 because if we add 2 cups of water to 2 cups of water, we will get 4 cups of water. Similarly, 2 bags of candies and 2 bags of candies makes 4 bags. The arithmetic rules are simply a reflection of how the physical world works. But what about adding 2 cups of water to 2 cups of pentanol? We will get 3.94 cups of the mixture, not 4 cups. Does that mean the arithmetic law should also include 2 + 2 = 3.94? Another example is the speed of light, we know that when measuring the speed of a human on a moving train, we will get the sum of the speed of the human’s running and the speed train as the result. But if we measure the speed of light on a moving train, we will still get the constant of the speed of light. Does that mean we should conclude that any number add 299 792 458, which is the speed of light in m/s, will give us 299 792 458? Of course not.
2 + 2 = 4 is not because adding 2 cups of water to 2 cups of water will get 4 cups of water, nor is it because 2 bags of candies and 2 bags of candies makes 4 bags. We can not use physical phenomenons as the reason why 2 + 2 = 4, because there are some physical phenomenons that match the arithmetical rules, as far as our experience has shown, and some simply don’t obey the arithmetical rules. I will not go into details of the proof here, 2 + 2 = 4 can be derived from the set of 6 primitive concepts and 3 relationships above, and no physical experience required. The reason that we can use 2 + 2 = 4 to calculate the result of adding cups of water together, is that just like the example of piles of oranges above, we can put those primitive concepts into context using definition by reference and test that those references of the 6 primitive concepts in the physical world satisfy the 3 relationships using the scientific method. The scientific method has verified that arithmetic is a suitable model to use to describe and predict the physical phenomenon of adding waters or combining oranges. For the water example, we can define 1 as a single cup of water, the set of all numbers as all the possible volumes of water by combining single cups of water, so 1 cup of water, 2 cups of water, 3 cups of water, and so on. Add can be defined as pouring two volumes of water together. By using these definitions, we can design experiments to test that pouring cups of water together satisfies the 3 relationships listed above, so we can use arithmetic to calculate the results of combining waters. But when combining water with pentanol, the second relationship is broken. If we define 1 as a single cup of liquid, either water or pentanol, adding two cups of liquid together will not result in a valid cup of liquid, when one of the cups is water and the other is pentanol. Remember, we define the set of all numbers as, 1, 2, 3, 4, etc., full number cups of liquid. Therefore arithmetic is not an accurate model for describing the phenomenon of mixing water and pentanol.
Science is the bridge between the physical world and mathematics. We use the scientific method to test if a given physical phenomenon shares the same relationships as the mathematical concepts we have defined, and which linguistic model can be best used to describe and predict each given phenomenon. We know that we can use arithmetic rules to calculate the number of oranges in a pile because we have used the scientific method again and again and verified that arithmetic rule has always been a good model to predict the number of oranges when we combining piles of oranges. While for other phenomenons such as mixing liquids or combining the speed of fast-moving objects in different frames of reference, arithmetic rules are only an approximation.
Even though mathematics is an intellectual construct defined in a vacuum, that is entirely separate from the physical world by design. But that does not mean that it is completely made up. As we have seen above, many physical phenomenons, from adding oranges to a pile to the light traveling through space, share the same relationships as our mathematical concepts. To be honest, we do not yet know why, objects in the physical world can, and do share the same relationships with pure intellectual constructs. This is one of the greatest mysteries of this fascinating world we live in. Mathematics is just like other languages, a way to help us understand the physical world and to communicate our understanding to others.
Thank you for listening to the podcast! If you like the podcast, please share it with your friends! I love asking difficult and interesting questions and I hope that you do as well.
RhaPsoDySea 