When we think of math, we think about equations, numbers and letters, the answer to a puzzle. Math helps us with our lives almost everyday, just in passing thought even. What we do not do however, is think about how math came to be in the first place. Did someone create it, let it come into their mind and release it for others to find out about, or was it founded, and people used their knowledge of it to create more ideas from it and has now become the universal language for all? In the best way to put it, mathematics was discovered. Mathematics has always naturally existed in nature and we just found a way to talk about it. Whether or not humans existed, there would have always been a concept of more and less and the relationships between. Math (as we are taught it) is just the common language we use to discuss and explain how to answer propositions.
Have you ever thought about the question “do trees exist or were they discovered?” It might be a stupid question to ask yourself, but when you think about, trees exist without a name without humans, but we did discover them and we did give them a name so we can discuss them. Math is the same way. Famous physicist Richard Feynman once said: ‘Not knowing mathematics is a severe limitation in understanding the world.” As much as some of us dislike math, it is an important part of society and our culture. Math is used all around the world almost the same way and is sometimes easier to retain than other subjects. You are constantly repeating the same things to help get you to memorize what you are learning and that can really help when it comes to studying and remembering information. We do not realize that we use basic math in our lives almost daily. It gives you a different perspective on how we learn things and is overall a helpful experience for everyone.
Since we are talking about mathematics being discovered, there is a big question a lot of philosophers have used, which is “do mathematical objects exist?” A question that can have universal answers because none of us know. Of course, in hindsight, mathematical objects do not prevail in the same way that a physical object prevail. You can physically touch a table, but you cannot physically touch a number. With this peculiar knowledge we can refer back to the seventeenth century, when Plato was thriving. Plato believed in rationalism, which is the theory that reason rather than experience is the foundation of certainty in knowledge. If you want to know something, you have to know how it works, rather than actually doing something to learn about it. This can refer back to the fact that mathematics was discovered and not invented because we learned theories of math and how they worked and applied it to everyday life which then created new and different ideas about math that spread throughout the world to where it is now. Immanuel Kant, another popular philosopher, had different ideas about mathematics. He believed in a priori which is “relating to or derived by reasoning from self-evident propositions.” He claims that mathematical judgments are always a priori judgment. They are necessary, meaning they cannot be borrowed from experience. Kant believed that it is unnecessary to subject mathematics to such a specific critique. Mainly because the use of pure reason in mathematics is kept to a “visible track” via instinct. His unique ideas about mathematical reasoning is also interesting because he has a claim that mathematical cognition comes from the “construction of its concepts.” So, for example, we all know what a triangle is. Kant believed that the concept of a triangle can be diffusely defined as a figure contained by three straight lines, meaning the triangle is “constructed.” Another example of how we can see that mathematics was discovered because we can take things in math that are super simple, such as shapes, and create new perceptions of the way we see them, rather than just looking at a triangle and seeing just a boring triangle. In a book called “Plato’s Ghost: The Modernist Transformation of Mathematics” by Jeremy Gray, Gray talks about Kant and his ideals about mathematics, stating “….mathematics yields theoretical knowledge purely, whereas physics is at least partially concerned with sources of knowledge other than reason” (Gray, 2008) [Pg. 78]. Kant believed that mathematics details to the forms of perception. In “Thinking about Mathematics: The Philosophy of Mathematics” by Stewart Shapiro, Shapiro talks about Kant and the way he sees mathematics in philosophy, stating “…mathematics is necessary and knowable a priori, but mathematics has something to do with the physical world” (Shapiro, 2000) [Pg. 23]. With this quote we can takeaway that mathematics relates to the way things happen around us physically, but are not physically tangible for us. Also bringing us back to the point that mathematics was discovered because of the physicality of it all.
In another realm of philosophy, we turn to Frege. Frege created foundations for contemporary discipline of logic. He did so by creating a system that allowed one to study inferences correctly, and also by creating an analysis of complicated sentences/phrases that showed a hidden unity of specific classes of assumptions. In “Thinking about Mathematics: The Philosophy of Mathematics”, Shapiro talks about Frege by stating that, ”…Frege held that only knowable or justifiable propositions can be analytic or a priori’ (Shapiro, 2000) [Pg. 109]. Meaning that Frege only thought that known or justifiable options could be a priori. Frege also believed in logicism, which is the theory that mathematics is an extension of logic, and therefore that some or all mathematics is reducible to logic. By this he is saying that all of mathematics, no matter the level, can be shortened to rationale.
Another philosopher, Kurt Gödel, believed that mathematics is involved with the regular features of the real world. Gödel seemed to claim that the axioms do not have to be obvious in what they are themselves. He believed that just like the way we study animals, we do the same with mathematical objects and their properties, and why they are so important. He also believed in intuitionism, which is the theory that primary truths and principles are known directly by intuition. Since intuition is the ability to understand something immediately, without the need for conscious reasoning, mathematics becomes inferior to this statement because before ever knowing anything about math, you need to learn about it to be able to know it off the top of your head.
Another argument would be Penelope Maddy’s naturalism belief, which is the idea or belief that only natural laws and forces operate in the world. In Shapiro’s “Thinking about Mathematics: The Philosophy of Mathematics”, he talks about Maddy’s ideals by stating she argues that “…onotological realism about a type of entity is justified of the objective existence of the entities is part of our best explanation of the world” (Shapiro, 2000) [Pg. 220]. This helps us in the argument of whether or not mathematics objects exist. Mathematics is not only essential to us in general terms but it is also essential to us for modern science. Physically, they may not be there, but from these arguments we can see that mathematical objects do exist, according to these philosophers.
Although these philosophers had good points on how mathematical objects exist, we turn to the other side for the philosophies that deny the existence of mathematical objects. We first turn to Hartry Field, who believed in fictionalism, which is statements that appear to be descriptions of the world should not be construed as such, but should be understood as events of ‘make believe’, of pretending to treat something as literally true. He agrees that math is effective in modern science, but in some sense, science can be done without the help of math. In a book called “The Indispensability of Mathematics” written by Mark Colyvan, he talks about fictionalism and gives examples on what it means. One of them being: “the mathematical fictionalist takes sentences such as “seven is prime” to be false (because there is no such entity as seven) but “true in the story of mathematics”” Colyvan, 2001) [Pg. 5]. With this belief Field can use it to his advantage and state that mathematical objects do not exist because of the pure fact that some things are false and are proven so.
Another anti-realist who has the same belief as Field is Charles Chihara. Chihara enforces linguistic items by talking about the fact that an open sentence is just a sentence in which one specific term has been replaced with variable. Just like numbers and sets, open sentences in an optimal development are abstract. Chihara said that real numbers were like open sentences because of the correspondence to natural numbers. In Shapiro’s Thinking about Mathematics: The Philosophy of Mathematics” he talks about Chihara’s ideas with open sentences and real numbers and states that “each axiom of real analysis, including the completeness principle, corresponds to a statement of which constructions are possible” (Shapiro, 2000) [Pg. 243]. With this explanation, we know that the applicable system of achievable open sentences to be constructed, would then be exemplified by the real number structure. This is an amazing thing to point out and reiterate that mathematics in philosophy is a universal topic. Many philosophers have different beliefs and because of this, give us many different sides to what we can talk about. With this, we can see that there are two sides to everything, but what about the middle ground philosophers? The ones that will say it depends, mathematical objects exist if…, what are their stances on this ubiquitous statement?
Bertrand Russell was a British philosopher in the twentieth century. He was a logician, meaning he believed in logicism. He agreed with Frege’s account of the natural numbers was right. Russell took his analysis of propositions, concepts and numbers and insisted on the vicious circle principle. This rejected the impredicative definitions. Russel believed that number is what is characteristic of numbers. Numbers was not a single number, but of some distinct number. In a journal called “Russell’s logicism” by Jeff Speaks, he talks about Russell’s ideals and how he believed numbers are, this was Russell’s example, “A trio of men, for example, is an instance of the number 3, and the number 3 is an instance of number; but the trio is not an instance of number” (Speaks, 2007) [Pg. 2]. This is a good lesson of showing that mathematical objects may exist, but only if it is attributed to a physical thing. Another philosopher who had middle ground ideas was David Hilbert. Hilbert believed in formalism, the study of a text without taking into account any outside influence. It is the view that mathematics is about characters or symbols, with the systems of numerals and other linguistic forms. Hilbert believed that the role of intuition and observation is explicitly limited to motivation and is heuristic. Because of these observations, if axioms were fulfilled, then anything at all can play the role of primitives. He incorporates these ideals into the existence of numbers by explaining that we learn on our own how numbers work and why they exist. We all come up with our own different concepts of the identity of numbers and how we use them. This argument helps with the ideas of showing that mathematical objects can exist only if we decide them to be based on our interpretations of it.
One last philosopher that also agrees with this concept is Arend Heyting who was a Dutch mathematician and logician. Heyting believed in truth conditions. That from the underlying metaphysical assumptions, realism is in truth value. It makes sense that Heyting holds the position we are mind-dependent when it comes to mathematical objects. We really are never able to grasp them on our own. Meanwhile, logicists and formalists have other ways of thinking that do not adhere to Heyting’s ways, by believing in only certain things that are not the same as what Heyting is telling us to believe. Logistics is generally the detailed organization and implementation of a complex operation, and a formalist is a person who adheres excessively to prescribed forms. So with Heyting’s belief in this, it proves that there can be multiple ideas in the solutions of believing mathematical objects exist.
Overall, one of the greatest things about philosophy is the fact that it is so comprehensive. Whether you agree with one side or the other, there is no right or wrong answer. Whether you believe mathematical objects exist or not, we are all allowed different answers to that statement. You may believe math was invented, which of course, from a general stance, does not seem to be true, but no one can tell you otherwise. Math will always be apart of our culture, it will stand the widths of time and with the discovery of it happening so long ago, it is no wonder we still use it today.