Mathematics and Reality

There are 2 main views about the relationship between mathematics and reality. One view states that we can know mathematical truths without any prior experience from reality. The other view states that we cannot know mathematical truths unless we have prior experience with reality. I will argue in favor of the second view. Proponents of the first view might advance an argument like this: “What “prior experience” goes into my knowledge that every natural number is the sum of four squares or the knowledge that every natural number can be uniquely factored into primes?”

Here is my response:

Math is ultimately dependent on direct experience with reality because (1) math is dependent on numbers, (2) numbers are concepts, and (3) concepts are ultimately dependent on perceiving reality.

Reality causes our sensations. A perception is a group of sensations automatically retained and integrated by the brain of a living organism. We simply perceive too many objects to remember every individual thing. We mitigate this problem through conceptualizing our perceptions. A concept is cognitive unit of meaning—a symbol that refers to objects in reality. Humans can form concepts because we can recognize similarities and differences among objects in reality. We summarize these similarities or differences through a process of abstraction.

Example: Green is a concept. Green does not exist by itself just floating somewhere in reality. There is no “greenness” that one can point to. But we can see a green mango, a green car, and a green turtle. These each have the property green. Green does not exist apart from green objects, but we can abstract the concept of green and talk about the concept of green independently of objects.

Numbers are the same way. There is no number 5 running naked in the wild. We derive the the concept of 5 by observing 5 fingers, or 5 mangos, or 5 turtles. From these experiences we can create the concept of “5 units”. We also observe that 2 units + 3 units = 5 units. Once we understand the number-concepts and how they integrate together, we can reapply mathematical concepts and rules back on the concepts themselves. Through this process we can gain knowledge without having further experiences. As Wittgenstein said, “From the given, I can construct what is not given.”

The “prior experiences” that lead to the knowledge that every natural number is the sum of four squares is not some experience about natural numbers and squares running around in the wild. The prior experiences were the perceptions in one's youth that led to the creation of concepts of numbers. The prior experiences also include hearing your teachers tell you how to integrate these number-concepts correctly.

This process of integrating concepts concepts directly is simply called reason. Reason is the ability to integrate concepts as derived from the senses. Our sensations are sensations about reality. Therefore, claiming that mathematics is an exercise in pure reason already presupposes necessary experience with reality.

Stereotypes

A stereotype is a conceptual categorization often in reference to a group of people. Examples of stereotypes are: Jews are wealthier than white anglo-saxton protestants, blacks are more likely to be on welfare than whites, students in business are more conservative than students in the arts, men are stronger than women, homosexuals are effeminate etc.

The most important question regarding stereotypes is this: Is it more often the case that stereotypes reflect reality or more often the case that they affect reality?

Those who assume that reality is mostly socially constructed tend to reject the idea that stereotypes reflect reality. Those who believe in an objective reality that exists independently of beliefs and opinions are more likely to be suspicious of the effect of stereotypes on reality. I find myself in this latter group for the following reasons:

First, the human mind forms concepts based on differences and similarities between objects, events, and people. Our ancestors in our evolutionary past had a stake in forming correct concepts about their world. Those ancestors who took the time to see if every large feline they met was dangerous would not have survived long enough to pass on their genes while those who were able to make quick judgements in order to react would have more likely survived. In other words, concept formation is useful because it helps us understand the world our around us without having to engage in the impossible task of treating every experience as completely new and wholly uncertain. I suspect that most people would find this uncontroversial except when it comes to concept formation (i.e., stereotyping) about people. 

The second and more convincing reason is this: the claim that human concept formation is merely socially constructed is self-refuting. An argument is self-refuting when it must assume the thing it is arguing against. The act of stating that something is true or false, or valid or invalid necessarily assumes the validity of concept creation. Therefore, rejecting the idea that stereotypes (in the sense of conceptual categorizations) in general reflect reality is also self-refuting. I qualify the previous sense with "in general" because I am not saying that all stereotypes are accurate. (That would be a false stereotype).

People don't hold stereotypes forever. As people live and engage with reality, people are often quite willing to change adjust their stereotypes. For example, my grandfather was a LA Firefighter during the Watts riots of 1965. Because of his harrowing experiences, my grandfather had formed some discriminatory feelings against blacks. However he also had many close associations with black co-workers for whom he had no negative feelings at all.

According to the cognitive scientist, Steven Pinker in his Pulitzer finalist book The Blank Slate,

"People's stereotypes are consistent with the statistics, and in many cases their bias is to underestimate the real differences between sexes or ethnic groups. This does not mean that stereotyped traits are unchangeable, of course, or that people think they are unchangeable, only that people perceive the traits fairly accurately at the time. Moreover, even when people believe that ethnic groups have characteristic traits, they are never mindless stereotypers who literally believe that each and every member of the group possesses those traits. People may think that Germans are, on average, more efficient than non-Germans, but no one believes that every last German is more efficient than every non-German. And people have no trouble overriding a stereotype when they have good information about an individual. Contrary to a common accusation, teachers’ impressions of their individual pupils are not contaminated by their stereotypes of race, gender, or socioeconomic status. The teachers’ impressions accurately reflect the pupil's performance as measured by objective tests."

As stated previously, accepting the general validity of stereotypes does not mean that all stereotypes are accurate. False stereotypes are likely to persist if there is no consequence for them persisting. And of course, stereotypes do not justify racism or sexism, though it does justify discrimination which is not a categorical evil. Stereotypes are not "simply human inventions that have done more harm than good." In fact it would be harmful to deny the realities reflected by stereotypes if our goal is to elevate cultures that disproportionately tend toward socially harmful behavior.

Check the comments for a good treatment of stereotypes in Steven Pinker's book The Blank Slate: The Modern Denial of Human Nature .

Do unicorns exist?: The limits of thinking

One can conceptualize based on what he/she can sense (see, hear, touch, etc.) in reality. Our minds have the capacity to create concepts that do not refer to anything in reality. We can do this by grouping objects based on similarities and differences. We can also combine concepts. Concept-combinations are the beginning of creativity. For example we can take the concept of horn based on our experiences with horns in the past. We can take the concept of horse based on our experiences with horses in the past. Then we can then combine them and create the concept unicorn.

We can do the same with the concept of horse and human.

Or with women and fish...

Mermaids don't exist in reality.

One might say, "but that is not a "real" mermaid." It is in the correction that they are making a mistake. There are no "real" mermaids or centaurs or unicorns in reality. They only exist as concepts derived from a certain combination of other concepts in the human mind. Science fiction writers like George Lucas who imagine aliens on other planets cannot really create a truly original alien. Imagined aliens are always combinations of what is seen on earth. They have human bodies with squid-heads or octopus tentacles with huge eyes etc.

Star Wars Aliens

Absolute "creativity" would consist of creating something out of nothing. This is impossible. The only meaningful way of talking about creativity is recognizing that creativity is only a process of combining what already exists in reality in new ways. We can create conceptual combinations of combinations of combinations.

This has an interesting implication. It means that the creativity of a generation is limited by the combinations of concepts developed by previous generations. Stories, technologies, and ideas in general are determined by which concepts were developed or not developed in the past. More on this in future posts.

Conclusion Premise 1: Thinking depends on concept-formation. Premise 2: Human concept-formation is limited by (1) the objects that we have access to in reality, and (2) by the higher lever concepts that we are aware of. Conclusion 1: Therefore thinking is limited by our awareness of objects in reality and higher level concepts. Conclusion 2: Therefore, you are only as creative as your conceptual resources. Conclusion 3: Therefore if you want to be more creative you must (1) consume large amounts of concepts (become an infovore), and (2) Exercise your mind by combining concepts into new ways.

Concepts and Reality

Sensations lead to perception. We simply perceive too many objects to remember every individual thing. We mitigate this problem through conceptualizing our perceptions. What is a concept? A concept is cognitive unit of meaning—a symbol that refers to objects in reality. Humans can form concepts because we can recognize similarities and differences among objects in reality. We summarize these similarities or differences through a process of abstraction. Let me give a few examples of concepts and concept-formation.

The color green: Green is a concept. Green does not exist by itself just floating somewhere in reality. There is no "green" that one can point to. But we can see a green mango, a green car, and a green turtle. These each have the property green. Green does not exist apart from green objects.

Colors: We can also create concepts of concepts. Once we understand the concept green and the concept yellow, orange, red, purple, and blue. We conceptualize these concepts into a higher concept called "color."

Properties: A property is an even higher concept. It is a concept that can group color, shape, mass etc.

Humans: When we see a particular human that we have never seen before, we don't think, "Whoa, I have never seen this before; I have no idea how to interact with it." Instead we automatically create the concept "human" through our past experiences with other humans and we then know how to interact with new "humans" that we have never seen before. Now the word "human" does not refer to any one thing in reality. There is no one "human" that we can point to. Instead "human" is a concept that refers to any object that shares certain similarities among other concepts such as arms, language, body odor, etc.

From these examples, we see that the human mind can create more concepts than objects in reality. Concepts can refer to reality. But the farther the concepts move away from reality, the more likely people will be confused. We must be vigilant in making sure that (1) concepts accurately reflect reality and that (2) our concepts are properly integrated with all other concepts to avoid contradictions.

An axiom is a foundational proposition that cannot be proved but can be validated. Propositions are axioms when one must assume the proposition when trying to deny it. Conceptualization is an axiom because one must use it in order to try and deny it. Likewise the validity of the senses are axiomatic because (1) the senses lead to concepts and (2) one must use concepts to attempt to reject the senses.