Here are some thoughts I had after finishing the Mathematics section in Theory of Knowledge.
Mathematics does appear to be an “island of stability” in an ocean of chaos; after all, mathematics appears to be the only area of knowledge that offers “definite” answers to questions. This quality of mathematics has caused it to become some sort of a universal language. In fact, the geometry that we study right now “is basically Euclidean geometry”, which was invented (or discovered, hm…) by Euclid, a Greek mathematician, in 300 BCE (van de Lagemaat 189).
Now, Euclidean geometry is a fine example of how mathematical systems function. It is referred to as an “axiomatic, deductive system” (handout), or alternatively, a formal system (van de Lagemaat 190). There are three fundamental elements to this mathematical system: axioms, deductive reasoning and theorems.
Axioms
Axioms are the fundamental assumptions that any mathematical system relies on; they are often thought of to be self-evident truths. However, the nature of axioms prevents one from trying to prove an axiom; one would be stuck in an “infinite regress” (van de Lagemaat 190) if this was attempted. Hence “obvious” knowledge is assumed to be true.
The four traditional requirements for an axiom are:
- Consistent (The same set of axioms should always produce the same answer)
- Independent (You should not deduce an axiom from another axiom; they are self-evident truths)
- Simple (Should be as clear as possible for easy comprehension)
- Fruitful (Have a diverse set of applications in terms of theorems produced)
(van de Lagemaat 190).
The fundamental problem with axioms lies in the assumption that axioms are actually true since there is not a method of verification besides “It’s obvious”. The knowledge issue that arises is, “To what extent are the assumptions behind axioms in a mathematical system invulnerable to error?” Since there are limits to the way of knowing, such as perception, reasoning and language, it is possible that the assumptions in axioms can be misrepresented and thus misused, making it vulnerable to error.
Euclid’s axioms were:
- It shall be possible to draw a straight line joining any two points.
- A finite straight line may be extended without limit in either direction.
- It shall be possible to draw a circle with a given centre and through a given point.
- All right angles are equal to one another.
- There is just one straight line through a given point which is parallel to a given line. (van de Lagemaat 190).
This critical knowledge issue was brought to the mind of mathematicians when some started to question the validity of Euclid’s axioms. What if they were not true, or were true in some cases? What if there were other forms of mathematical “truth”? Prior to the 19th century, Euclidean geometry was assumed to be correct because the axioms were true since they passed both truth tests: they were logically consistent and true in the world (handout). After all, how could one doubt “self-evident truths”?
Challenges to Euclidean geometry came from Carl Friedrich Gauss, Nikolai Lobachevsky and Bernhard Riemann. All of these alternatives accepted Euclid’s first four axioms because they were extremely obvious; there were no visible alternatives. This illustrates the relationship with perception and mathematics: is there such a thing as “universal” perception in mathematics? Among the alternatives mentioned previously, the most popular is Riemannian geometry. His system of geometry challenged those who believed that non-Euclidean axioms would “lead to a contradiction and so collapse” (van de Lagemaat 205). However, no contradictions appeared, even though he had replaced Euclid’s first, second and fifth axioms. Furthermore, Riemann was able to modify these axioms by assuming that space was like the surface of a sphere, not an infinite plane that Euclid assumed. Essentially, Riemann proved that there can be multiple approaches to describing physical reality, and that axioms were not invulnerable to criticism and interpretation.
However, Riemann’s system also raised the problem of consistency: how could one prove that a mathematical system is free from contradiction? Can we truly accept a mathematical system as being accurate? Hence Kurt Godel came up with Godel’s Incompleteness Theorem, which stated that it is impossible to prove that a formal mathematical proof is free from contradiction (van de Lagemaat 208). In other words, we cannot be certain whether or not mathematics is free from contradictions; it cannot give us certainty.
Riemannian geometry also introduced the idea that formal systems have limits. For example, Euclidean geometry would appear to work quite well for regular life, but it is extremely poor for understanding flight plans and space around stellar phenomena, such as black holes. In essence, it would appear that there are multiple approaches to mathematical truth, but as illustrated by Godel’s theorem, we cannot know whether or not these approaches contain contradictions which can compromise their intellectual integrity.
Deductive Reasoning
This is essentially the application of deductive reasoning on axioms. For instance:
All U.S. Presidents were American citizens.
Abraham Lincoln was a U.S. President.
Therefore Abraham Lincoln was an American citizen.
Theorems
Theorems are concepts and “rules” that arise when one applies the axioms of a formal system onto an example. Once established, theorems can be used to create even more theorems.
Example: p191 textbook
Mathematical Community
In the mathematical community, new studies on mathematical discoveries are often peer-reviewed in journals since it is often thought that other mathematicians should be able to detect flaws in reasoning or application of mathematical concepts. The question, to what extent does peer review stifle or encourage academic innovation? The mathematical community is limited by its interpretation of language, perception, reason and in some cases, emotion. The underlying assumption in the mathematical community is that those with mathematical experience are more intelligent than the researcher, and hence have authority. However, as we have seen in the example of Euclidean geometry, is it not possible that a mathematical paper could base its assumptions on a different set of axioms? Or be useful at a later time and date? Thus it is possible that the mathematical community may falsely assume something to be false when it is in reality true. On the other hand, it is also possible that peer review can help prevent the mathematical community from being inundated with papers that are extremely inaccurate and misleading that could suppress mathematical research progress. Thus, peer review does have pros and cons, but its integral nature in the mathematical community does seem to suggest that the pros outweigh the cons.
Furthermore, the competitive nature of the mathematical community often generates extreme competition and the idea that quantity matters more than quality. For instance, in China, some researchers use Google Scholar to find American scientific research in order to find data or analysis for their own paper. Not only does this compromise mathematical research, but it also causes problems for this area of knowledge, since it might adversely affect the axioms, deductive reasoning and theorems of a formal mathematical system when quality is compromised for quantity.
Intense Cogitation relies on your donations! Donate today, whether it be $1 or $5... you choose!
I would argue that in the realm theory of knowledge, Mathematics (well, frankly, deductive reasoning in general) acts less as a stabilizer and more as a sore thumb, for lack of a better word at this time. Essentially, it removes clarity from the notion of “knowing” by creating the illusion of reason based on absolutes.
In the realm of Mathematics, axiomatic truths seem self-evident based on an observational understanding of perfect shapes, in the case of Euclidean geometry. It just makes sense that through any set of two points can be drawn only one line (or several overlapping lines, as it were, but those don’t count). It also just makes sense that through any three non-collinear points there exists only one plane to connect them. And from there, it seems logical to conclude that any two lines that intersect at one point create only one plane (essentially, if the intersection point acts as one point, and one point on either of the two intersecting lines act as respective points, we have three non-collinear points, which is the basis for a distinct plane. It just makes sense).
This creates problems in the realm of understanding what it is to know, though, because it sets a precedent of prerequisites for truth. It also characterizes truth as having a basis of logical conclusions derived from more self-evident truth. See, axioms, postulates, and theorems work very well for things like Euclidean geometry for the simple reason that geometry is perfect. It rests on the foundation that there are perfect shapes and structures, and it derives meaning and purpose from the assumption of perfection (kind of like basic physics, in a way..).
This begs the question: can one know something that cannot rightly exist? Basically, is mathematics knowledge if its basis is unattainable? Is its basis unattainable? On one hand, mathematics can emulate imperfect situations once the proper corrections are made for things that do exist. However, does that mean it is right, or simply that it is a tool to approach rightness.
That said, this by no means means Mathematics is not legitimate way of knowing (or more aptly, way of understanding). It does, however, pose some clearly identifiable harms by virtue of the impression it creates of knowledge. When deductive reasoning is used in the definition of knowledge, knowledge is unattainable (JTB):
The Justified True Belief (JTB) Definition of Knowledge (courtesy of Plato, or perchance one of his students)
S knows that proposition P is true if and only if:
1. P is true
2. S believes P to be true
3. S is justified in believing P is true
But then comes along thought experiments like the clock riddle, and one things leads to another, which leads to Descartes, which leads to “we cannot know something to be true”. And philosophy explodes.
So, we are left with the conundrum that truth a) can’t exist, b) can’t be known, or c) is approached (perhaps much like limits in Reimannian geometry) in ways we can begin to understand, though perhaps never conclude with absolute certainty, or d) [insert legitimate option here?].
Funny, it seems like math has gone through a lot of the same issues about certainty that philosophers have, just with a lot more numbers and a lot less consideration of moral relativism.
But, in line with my original purpose, Mathematics, by virtue of its basis in deductive reasoning, creates a highly simplified image of knowledge that, by being an “island of stability” actually creates false perceptions of what it is to know. However, its simplicity in that regard can act as a stabilizing factor when you consider the complexity of the issue of knowledge in and of itself. Its simplicity perhaps allows for a better initial grasp of knowledge issues and concepts, and once one understands where confusion arises in the Mathematical discipline, one can then start to try and understand the shortcomings through other ways of knowing, or other areas of knowledge (religion makes for a particularly interesting one, though highly complex for analysis).
Also, Brian: I am impressed. This site is pretty damn cool.
Hey Brian, what’s your last name? I’m citing you for my tok essay. reply asap! my essay due tonight
For sourcing, you usually can just use the website name.
Dunno if your TOK teacher will accept a blog as a source though–you should double check to make sure.