The mathematics of thought is something that interests me very much in my research. How does mathematics shape our conceptualization of reality? That is the basic question that I always ask myself. You might have a trivial answer to this question because mathematical methods are everywhere, especially in the empirical sciences such as physics.
Physical theories cannot be formulated without using mathematics. In biology, we use mathematical models to study the behavior of specific patterns of organisms, and in sociology, we use mathematics to analyze social networks. In that sense, mathematical methods are everywhere in empirical science because empirical science has shaped the way we see reality. Naturally, mathematics has a part in that as an indispensable scientific tool.
However, I am more interested in mathematics as a philosophical paradigm. To give you an idea of what that means, I’m going to introduce five examples from the history of philosophy.
The first one, of course, is Plato, who thought that mathematical knowledge was the highest form of intelligence. We know this because he had engraved the sentence, “Let no one ignorant of geometry enter here!” on top of the entrance to his Academy. He always believed that proficiency in mathematics was a prerequisite for the acquisition of any further knowledge.
Galileo had a slightly different view than Plato. He believed that mathematics was the language of the book of nature. He thought that it was the scientists’ task to become proficient in that language and thereby study the fundamental structures of reality.
Although Kant had an extraordinary place in his philosophy for mathematics, he thought that mathematics is a prime example of a particular class of human judgments. He called judgments as synthetic a priori. We formed those judgments without consulting our experience, but still, they were analytical judgments, and we got to learn something new in those judgments. According to Kant, we could learn how these judgments work in different contexts.
Nietzsche believed that mathematics was a paradigm of refinement and rigor. He thought that all other sciences should strive to be as much as possible, like mathematics, in that respect.
Lastly, one of the most influential mathematicians of all time, Cantor, who proved that there are infinities of different sizes. He found very straightforward and simple proof that the set of real numbers is larger than the set of natural numbers even though both are infinitely large. He also called these infinities transfinite numbers. Then he came with an idea of absolute infinite that was a manifestation of God. In other words, Cantor found something divine in mathematics.
To give you a slightly better picture of the relationship between mathematics and the empirical sciences and how this relationship has evolved, we should go back to ancient Greek. For the ancient Greeks, mathematicians such as Plato, Pythagoras, and Euclid were the only source of true knowledge of certain knowledge. In contrast, our beliefs about the empirical world were considered to be mere opinions. It is because the empirical world is continuously changing. He believed that it’s impossible to have specific knowledge about something ever-changing.
We see here that mathematics is the sort of knowledge above all, and the empirical world is something we can have opinions about but not know anything about — a very distinct separation between the two realms.
When empirical scientists started to blossom, and the exploration of the empirical world started to become more and more systematic, this picture changed dramatically during the times of the Scientific Revolution many centuries later. Today, we’ve arrived at a picture where mathematics was considered the language of nature. Scientists were considered to learn that language and also figure out the fundamental laws of the universe. These laws were thought to be mathematical laws. So by putting mathematical laws with our observations, the scientists were supposed to understand the fundamental truths of reality. This picture changed again in the 19th century, the time that we now call empiricism. During that time, the fortunes of mathematics were completely reversed again. For example, some empiricists believed that mathematics didn’t have any content of its own. Yes, it is a handy tool for the empirical sciences but a mere tool nevertheless. The only source of certain well-grounded knowledge was considered to be the empirical sciences.
So mathematics started as being this highest form of knowledge. Then it is merged with the empirical sciences. Later the empirical sciences became our paradigm of well-grounded education, and mathematics was considered to be a mere tool for the sciences. Another way of saying this is mathematics started as something that could be called the language of God. The ancient Greeks believed that the reason that we find sort of imperfect geometrical figures in the empirical world is because the gods gave geometry to the world to make us understand that this nature is an intelligible environment.
After the scientific revolution, mathematics became the language of nature. During empiricist times, it had become or taken a fall down to the level of a mere language of men. What we see here is the continuous dissociation of mathematics from the physical world. The crossover point that I mentioned earlier coincides roughly with the rise of pure mathematics.
So in the 19th century, mathematical concepts had little or no physical meaning started to be introduced by mathematicians as legitimate objects of mathematical study such as dimensional spaces, transfinite numbers, non-euclidean geometries, complex numbers. All these mathematical objects did not have any definite physical meaning anymore. So mathematics, in that sense, moved beyond concepts that were suggested directly by our experience of physical reality.
At the beginning of the 20th century, Morris Klein was one of the most essential and well-known historians of mathematics. Once he writes very nicely, the circle within which mathematical studies appeared to be enclosed at the beginning of the 19th century was broken at all points. Mathematics exploded into a hundred branches that each built on its system of axioms.”
So that’s a very vivid picture of what happened to mathematics. Exploded into all these different subfields and people working in these various subfields wouldn’t necessarily talk much to people working in other mathematical subfields.
In the first half of the 20th century, set theory, one branch of mathematics, suddenly acquired a lot of importance. It was because a group of French mathematicians was working under the sort of pen name Bourbaki group started to prove that set theory is a foundation for all of mathematics. They showed that we could express all mathematical theorems in the language of set theory, which of course, made visible the interconnections between all these branches of mathematics. It made evident the fundamental assumptions that were being made in all the various branches of mathematics. Most importantly, it provided a unified framework and a unified language in which all the different branches of mathematics could be interpreted.
The set theory created a foundation for all mathematics, but it did more than that. It created an autonomous mathematical paradise. That is David Hilbert’s expression; “As such as this freestanding paradise, this freestanding mathematical realm became a perfect model for other domains, at least from the philosopher’s point of view.”
So just to summarize this again to make the movement visible.
- In the beginning, ancient Greece, ultimate reality is where mathematics is in the Platonic heaven of eternal forms.
- During the times of the Scientific Revolution, the ultimate truths of reality are actually hidden in nature.
- During the time of the empiricists, mathematics has become nothing but a mere tool.
The only reality we have is this physical reality, but actually, a mathematical reality and an empirical reality coexist. They exist independently of one another, but they intersect in the area of Applied Mathematics. I want to point out to you that the perception of reality changes parallel to the movements in the history of mathematics.
A mathematical ideal does not only shape our perception of physical reality, but it also shapes how we understand and conceptualize other very fundamental areas of our lives. Mathematics is ideal because mathematical truths are indisputably certain. Mathematical knowledge gives us 100% certainty. Mathematical statements have objective and determinate truth values.
In that sense, mathematics has become a benchmark for all other areas of discourse that have a claim to objectivity. That makes mathematics very interesting to philosophers as a model for other non-empirical domains. What philosophers tried to achieve is similar to mathematics because that generates credibility. They think that they can draw interesting inferences from comparing mathematics to other areas such as ethics, logic, metaphysics, and religion. You might think that what do these domains have in common with one another.
Look at the statements of these domains.
Look at the statements of these domains.
In maths, we have statements like two plus two equals four,
In ethics a statement like torturing babies for fun is wrong,
In logic, a statement like if p then q,
In metaphysics, there could be purple pigs,
And in religion, we say if you’re religious, God created the universe.
In terms of their actual content, these statements do not have anything in common, of course. They talk about very different things, but from a philosophical point of view, they are related. The reason is that, first of all, they refer to non-empirical entities that are beyond the grasp of empirical investigation such as numbers, values, norms, reasons, moral properties, propositions, possible worlds. All these things are things we cannot investigate with our traditional means of empirical science.
Many of us would intuitively think that these statements are true. So that’s a different way of saying that many of us are realists about these domains. What does that mean to be a realist? First, it means that we’re in some sense committed to their being non-empirical entities like numbers, moral entities, values, etc. It also means that we believe in statements in those domains having objective truth values. We also believe that the truth of those domains is not reducible to truths about something else.
We believe that these truths are knowable. Thus we can get to know mathematical, ethical, metaphysical truths, etc. If you’re a realist about these domains and if you’re committed to at least some of these claims, you’re also going to face several challenges.
I want to give you some quotes, so you can see that this is an actual debate going on by actual people.
Moral anti-realists would say stuff like this; “Radical differences between first-order moral judgments make it difficult to treat those judgments as apprehensions of objective truths.” Brian Leiter then supports the point and argues that “Persistent disagreement on foundational questions… distinguishes moral theory from inquiry in the sciences and mathematics.”
So what they’re pointing out is that mathematicians do disagree radically and passionately about several things most notably about questions of mathematical foundations.