Tag Archives: Philosophy

Faith and Math: the Origins of Math

But to us there is but one God, plus or minus one. —1 Corinthians 8:6±2.

Religions. Image: Randall Munroe.

Religion and math are oft thought of as being separate and often in opposition, at least within western society. We recently learned about the connections between math and religion in India (http://www.bbc.co.uk/programmes/b03c2zvr) but did not explore where else faith has had an impact on mathematics.

Where does math come from?

The main two answers to this are as follows: humans discover math, or humans create math. In the case of the first, it is accepted that all of math exists, has existed, and will always exist, regardless of whether or not we are aware of it. Even though the ancient Greeks were unfamiliar with negative numbers, negative numbers existed, but had simply yet to be discovered. This mode of thought is described as mathematical realism, and can be defined as the belief that our mathematical theories are describing at least some part of the real world (http://web.calstatela.edu/faculty/mbalagu/papers/Realism%20and%20Anti-Realism%20in%20Mathematics.pdf pg. 36). There are several subdivisions among this group and more detail is given to this later. The second statement, that humans create math, is characteristic of mathematical anti-realism. By this mode of thought, math does not necessarily have any connection to the real world; it exists because we create it and it is true because we have made it to be true.

The Realists

The realists should probably be subdivided into two main groups: Platonists and everyone else, with the “everyone else” being a minority, so we should probably have a definition for mathematical Platonism. According to both Stanford’s and the internet’s encyclopedias of philosophy, mathematical Platonism is based on the following theses: Existence, Abstractness, and Independence. Basically, mathematical objects exist they are also abstract, and your language, thoughts, religion, or anything else doesn’t change what they are.  I should probably also mention that there are also subcategories amongst the Platonists, like traditional Platonism, full blooded Platonism, and some others, but I don’t want to get into that.  There are, however, mathematical realists who do not subscribe to Platonism. One such group is the physicalists. A strong proponent of physicalism was John Stuart Mill. The argument for this is that math is the study of ordinary physical objects and is therefore an empirical science. According to this mathematics is basically meant to discover laws that apply to all physical objects. For instance, 1+1=2 gives us the law of all physical objects that when you have 1 of the object and you and another of the object you have 2 of the object instead. This differs from Platonism in that these objects are no longer abstract, but rather describe all objects. These are not the only two categories of realists. The main problem I have with this is that it means if all objects were to vanish math would cease to be true. This is because physicalism is not based on the abstractness of mathematical objects which means that the objects themselves must exist.

The Anti-realists

Anti-realism is in general the belief that Math does not have an ontology. As with mathematical realism there are a lot of different subcategories of mathematical anti-realism. I’ve chosen to talk a bit more about conventionalism and fictionalism because they seemed interesting.

Conventionalism holds that mathematical statements are true only because of the very definitions of the statements. By this mode of thought, math does not necessarily have any connection to the real world; it exists because we create it and it is true because we have made it to be true. The statement “pi is the ratio between the circumference of a circle and its diameter” is true only because we define a circle as being a shape with a radius r, a diameter 2r, and a circumference 2*pi*r, and not because the universe made it so. In this sense, the above statement makes about as much sense as “all bachelors are unmarried”; both are obviously true, however this is because of their definitions rather than being the result of some universal laws.

Fictionalism argues that statements like 1+1=2 make about as much sense as “Harry Potter’s owl was Hedwig”. Yeah it’s true, but only within its given context. It is important to note that statements such as 1+1=3 make about as much sense as “You’re NOT a wizard, Harry”, because given the context of the story, or fiction, these statements simply make no sense. There are some interesting similarities about Fictionalism and Platonism. The biggest one is that both of them take mathematical statements at face value. This is to say that both of them take 1+1=2 to mean that to add the mathematical object 1 and adding it to another mathematical object 1 will result in the mathematical object 2. The difference then is that where Platonism takes this to also mean that these abstract objects exist, Fictionalism does not accept that these objects exist. This is different from conventionalism in that conventionalism doesn’t even accept that you are referring to objects, regardless of their existence. The thing about Fictionalism is that the subject doesn’t technically actually even a little exist. By this I mean that Harry Potter doesn’t actually exist (probably) and that therefore he isn’t actually a wizard (probably) and that since he doesn’t exist he doesn’t actually own an owl named Hedwig (probably), and that by that same logic 1 doesn’t actually exist, and neither does 2, and 1+1 doesn’t equal 2 because none of them exist.

Implications of these schools of thought

Mathematical realism, in a certain sense, seeks to prove truths about the universe. This is most obvious when you consider modes of thought like physicalism, under which math would be a really general science, but even under Platonism you are seeking to find laws that govern these abstract objects you are finding. So for instance, when you have one of some object, and you add another of that object to that first object, you now have two of that object and according to mathematical realists, this is true. It is a fact. According to mathematical anti-realists, if you remove the humans, or whatever it is that is observing this addition, then there is no longer a group, one of the things, or two of the things. These concepts existed only because the humans said they existed, and when the humans stopped existing and thus stopped observing this these things lost the properties of being one, being grouped, and finally being two. The exact way in which this is argued depends on what subcategory one subscribes to. (https://www.youtube.com/watch?v=TbNymweHW4E&list=UU3LqW4ijMoENQ2Wv17ZrFJA)

How this relates to faith

Regardless of whether you believe that the statement “pi is the ratio between the circumference of a circle and its diameter” is true because of universal laws or because of human created definitions, the statement is still true. The importance of this is that it means that there, at least at this point in time, is no way to verify whether the reason for math existing is tied to the very nature of the universe or whether it is simply the product of the human mind. As a result of this, the belief in either of these theories is, at least in a certain sense, a leap of faith.

My thoughts on this

My personal opinion on this leans towards mathematical realism and more specifically Platonism. I agree that mathematical objects exist, but that they do not by necessity have a real world counterpart and thus are abstract, and I believe that regardless of whether or not humans exist, the mathematical concepts we have found to be true will still be true, even if no-one is around to appreciate, understand, or use them. One big reason I have for thinking this way is because of how various isolated cultures ended up discovering the same mathematical principles. By this I mean that counting systems, simplistic though they may have been, were not a unique event to just one area, but rather a common feature. I mean the Mayans had a counting system, so did the Greeks, Egyptians, Babylonians, Indians, etc. It seems somewhat unlikely to me that all these isolated cultures would create a method for defining something that doesn’t exist.

Additional reading/sources

Idea channel’s episode titled “Is Math a Feature of the Universe or a Feature of Human Creation?”

https://www.youtube.com/watch?v=TbNymweHW4E&list=UU3LqW4ijMoENQ2Wv17ZrFJA

Mark Balaguer’s “Realism and Anti-Realism in Mathematics”

http://web.calstatela.edu/faculty/mbalagu/papers/Realism%20and%20Anti-Realism%20in%20Mathematics.pdf

Stanford’s Encyclopedia of philosophy entry on Platonism and mathematics

http://plato.stanford.edu/entries/platonism-mathematics/

The Internet’s Encyclopedia of philosophy entry on Platonism and mathematics

http://www.iep.utm.edu/mathplat/#H1

Wikipedia’s entry on philosophy of mathematics. No, this was not used as a source; it is however, useful for additional reading.

http://en.wikipedia.org/wiki/Philosophy_of_mathematics

Mayans count as well

http://maya.nmai.si.edu/maya-sun/maya-math-game

Greeks count as well

A History of Mathematics, Merzbach and Boyer, pages 52-55

For Babylonian counting see

Plimpton 322

For Indians having a number system click the bbc  thing below

For the link to the bbc story thing

http://www.bbc.co.uk/programmes/b03c2zvr

for the comic

http://xkcd.com/900/

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Is Math Culturally Independent?

Is Math culturally independent?   Eleanor Robson asked this question regarding Plimpton 322. She wrote, “We tend to think of mathematics as relatively culture-free; i.e., as something that is out there, waiting to be discovered, rather than a set of socially agreed conventions.  If a simple triangle can vary so much from culture to culture, though, what hope have we in relying on our modern mathematical sensibilities to interpret more complex ancient mathematics?”  And yes, this was a homework question, but for some reason this question stuck with me, and I went looking a bit further.  For those of you who may not know, Plimpton 322 is an ancient Mesopotamian tablet around which there is some controversy. ¹  Scholars have claimed that Plimpton 322 is anything from a set of Pythagorean Triples, to a table of reciprocal numbers, or that it is possibly a trigonometry table.  The truth is, we just don’t know for sure; but whatever it is, it is definitely  more complex than the tax forms or accounting forms we typically expect the Mesopotamians to have left lying around.  (I just put it down a second ago, where did it go?)  Robson’s comment about the triangle mentioned refers to the difference between how we normally picture or represent a triangle and the standard Mesopotamian way of representing a triangle.  We have a tendency to depict a flat side facing down (for example Δ). The Mesopotamians, however, tended to represent their triangles pointing to the right similar to our play symbol. (Emblem unavailable at this time. Please consult your mp3 or video player, sorry for any inconvenience).

This question from Robson brought to my mind the idea of Musica Universalis.  Musica Universalis2 is a philosophical concept that is based on some assumptions made by the Pythagoreans, namely the combination of math and theology.  The Pythagoreans belived that everything had a numerical attribute,³ and they also found an appeal in certain symbols, such as the tetraktys and the Harmony of the Spheres (another name for the Musica Universalis).4  The concept of Music of the Spheres concerns the movements of the Planets, the Stars, the Moon and the Sun. (Remember, the thought at this time was that they all revolved around the Earth.)  One way of interpreting this was that there was some vast Celestial Orrery or Machine that had been set into motion. This Orrery controlled not only the motion of the celestial bodies but also the affairs of men. During these millennia there was no distinction between astronomy and astrology.

An example of an orrery. Image: Sage Ross, via Wikimedia Commons.

An example of an orrery. Image: Sage Ross, via Wikimedia Commons.

Johannes Kepler is a well know and still revered astronomer.  Kepler also believed there to be no distinction (at least it is not recorded) between astronomy and astrology and as an adviser and astronomer to Emperor Rudolph II he made horoscopes for not only the Emperor but also various allies and foreign leaders.  Johannes Kepler believed he had worked out much of the celestial orrery in his Mysterium Cosmographicum.5  The commonly held belief of the time was that all things could be understood by observing natural motions; whether those motions were of the planets, the stars, or in some cases the patterns of other natural phenomena.

Since all patterns can be represented mathematically, math then becomes the language of the universe. This idea can also be traced back to the Pythagoreans.  The concept that everything is a piece of celestial machinery that can be understood through math is still around us to this day, or at least it seems that the repercussions of it are. After all, if everything is patterns, and patterns can be interpreted wonderfully though Math, then Math must therefore be the Language of the universe. (That’s logical, that is.)  This seems to be the idea that Eleanor Robson is arguing against.  (Frankly, I agree with her.) This concept of a pure language of math is rather a strange convention that our society has if you really think about it. After all, the argument could be made that English (or any language really) is some sort of divine language because we can use it to so eloquently describe the world around us. Or perhaps Music is our divine language. It is pattern based, after all. So is this idea of everything being describable through math a belief we have found to be true, is it a truth that we somehow stumbled upon millennia ago, or is it a conceit of our culture?

1 A History of Mathematics by Uta C. Merzbach & Carl B. Boyer

2 Musica Universalis, Wikipedia

3 http://www.math.tamu.edu/~dallen/history/pythag/pythag.html

4 http://www.britannica.com/EBchecked/topic/485235/Pythagoreanism

5 Johannes Keppler Wikipedia

6 http://www.crystalinks.com/musicspheres.html

Archetypes of Wisdom: An Introduction to Philosophy , Douglas J. Soccio