Tag Archives: realism

Math: Is It All In Our Head?

After years of math classes, the crazy truth is finally coming out.  It is all just in our heads.  No way! How can that be? There’s an interesting debate in the world of math.  Are math principles the creation of humanity, or are they universal truths that humans discovered? There are compelling arguments on both sides of the debate and both sides have several different sub-levels of thought.  In this article, I will discuss them both generally.

The realists maintain that mathematical principles would exist even without people.  Humans discovered the principles and brought them into practical use and any intelligent human being could also discover the same principles.  This argument is supported by the fact that many cultures have discovered mathematical principles independent of one another.  Also, mathematical concepts, such as the Fibonacci sequence and some fractals, occur in nature which would suggest that they exist even without people.   Some realists, like the Pythagoreans, believe that the world was created by numbers.  The realist point of view can lead to an almost supernatural view of mathematics.

The challenge with mathematical realism is that there is no physical domain where math entities exist.  We cannot draw a perfect circle or even a line.  We can conceptualize these things in our mind and we can prove them in theory; however, we cannot actually manipulate math entities in the physical world.  Many math concepts exist only in the context of our understanding about them and conceptualizing them.

Another view is the anti-realists.  They maintain that math is the creation of humans in order to make sense of the world.  They recognize that math is an amazing, complex system and that it works as modeled by science.  However, some argue that scientific principles could be explained without math.  One anti-realist, Hartry Field, demonstrated this by explaining Newton Mechanics without referencing numbers or functions.  He explained that, in his opinion, math is fictional and is true only in the context of the story in which it is being told.

So, is it all in our heads?  A fiction that was created to explain properties in our world?  In reality we may never be able to settle the debate and it may not matter.  Math works.  That is the beauty of it.    In his article “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”, Eugene Wigner observes that

the mathematical formulation of the physicist’s often crude experience leads in an uncanny number of cases to an amazingly accurate description of a large class of phenomena.  This shows that the mathematical language has more to commend it than being the only language which we can speak; it shows that it is, in a very real sense, the correct language.

Albert Einstein, 1921. Image: Public domain, via Wikimedia Commons.

Albert Einstein, 1921. Image: Public domain, via Wikimedia Commons.

Perhaps the best thing to commend mathematics as being real, is that it works.  Time and time again, it works.  Its principles, laws and theorems, applied over and over, in different settings produce accurate results and predictions.  Einstein commented in a 1921 address titled Geometry and Experience, “It is mathematics which affords the exact natural sciences a certain measure of security, to which without mathematics they could not attain.”  He explored the question of how math, a product of our mind can be so applicable to the concrete world.  He asked, “How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirable appropriate to the object of reality?”  Einstein answers this question with the statement, “As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.”  He looks specifically at the field of geometry and the need humans have to learn about the relationships of real things to one another.  Even though the axioms of geometry are based on “free creations of the human mind”,  he says, “Solid bodies are related, with respect to their possible dispositions, as are bodies on Euclidean geometry of three dimensions.  Then the propositions of Euclid contain affirmations as to the relations of practically-rigid bodies.”  The abstract principles, when applied to “real” world situations prove to be accurate.  Einstein continues to explain how the theory of relativity rests on the concepts of Euclidian and non-Euclidian geometry.  He challenges the mind to conceptualize a universe which is “finite, yet unbounded”.  In the end, it is this ability to use conceptualized principles and apply them to our world that makes mathematics work.  So yes, mathematics may be all in our head and it may be a huge puzzle created by humanity, but it is effective, useful, and even beautiful.

Sources

Einstein, Albert. Geometry and Experience. http://www-history.mcs.st-andrews.ac.uk/Extras/Einstein_geometry.html

Wigner, Eugene. The Unreasonable Effectiveness of Mathematics in the Natural Sciences. http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html

Wikipedia. Philosophy of Mathematics. http://en.wikipedia.org/wiki/Philosophy_of_mathematics

Advertisements

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/