A few weeks ago my daughter sent me a Calvin and Hobbes cartoon. The dialogue went like this:

“You know, I don’t think math is a science. I think it’s a religion.”

“A religion?”

“Yeah. All these equations are like miracles. You take two numbers and when you add them they magically become one new number. No one can say how it happens. You either believe it or you don’t. The whole book is full of things that have to be accepted on faith. It’s a religion!”

“And in the public schools no less. Call a lawyer.”

“As a math atheist, I should be excused from this.”

At first, it is funny because it lampoons such a badly misguided attitude. Mathematics is the opposite of revealed religion because you are required to take nothing on faith: something only becomes a mathematical truth when a proof is found, and each individual mathematician is responsible for following the proof to the point of becoming convinced of the truth. The only major religion which seems in any way similar is Buddhism, whose founder told questioners (perplexed about the multiplicity of religious teachers at the time) to examine what they said; accept what is good, and reject what is not good.

One can see shadows of the attitude among some students, simply because mathematics to them is a complete mystery. Lacking the attitude that they should only accept it if the proof is convincing (most likely, like Calvin, their teachers never told them this), it all becomes rather mysterious, and there is a human tendency to put the mysterious, the supernatural, and the religious in the same mental box.

But perhaps, like all good jokes, there is some element of truth in it. Perhaps it is something like this. Mathematicians create their own mental universes, after all; maybe there is not so much difference from an algebraist beginning a lecture with “Let *G* be a group” and the God of Genesis saying “Let there be light”. The second statement, we are told, called light into existence in the real world; the first calls a group into existence in the mental universes of the lecturer and audience.

Mathematicians, for example, are one group of people who are (mostly) quite comfortable with the notion of infinity.

A few years ago the BBC World Service made a programme about infinity, to which I made a small contribution (a half-hour interview was reduced to three short soundbites in the final programme). After my last appearance, the psychiatrist Raj Persaud came on to explain that in his clinic he sees many people who have gone mad thinking about infinity; the clear implication was that the crazy mathematician who has just been talking is likely to be one.

But in fact most mathematicians work with the infinite every day, and as far as I know, we don’t have a higher rate of madness than any other profession. (It is true that Georg Cantor, who devised the theory of infinite numbers we use now, went mad; but this was perhaps partly because implacable opposition to his ideas from conservative mathematicians such as Kronecker essentially destroyed his career.) We devise ways of thinking about, and making mental pictures of, infinite sets in the same way that we do for numbers or groups.

An excellent recent book, *Naming Infinity*, by Loren Graham and Jean-Michel Kantor, describes the founding of descriptive set theory in the first half of the twentieth century. Part of their story revolves around the fact that, while the French mathematicians Borel, Lebesgue and Baire drew back from the implications of their discoveries, the Russians Egorov and Lusin pressed boldly on (even despite ideological opposition to their views from Stalin’s regime, which caused them enormous hardship). Their boldness sprang in part from the influence on them of Pavel Florensky, who was both a mathematician and an Orthodox monk, a leader in the “Name worshipping” movement in the Russian Orthodox Church. They were not afraid to name God, and likewise to name infinity; and once named, it could become the subject of mathematical analysis.

Does mathematics combine the mystical and the rational, as no other subject can? Maybe.

Or maybe not. I had a student in Oxford who read Mathematics and Philosophy because he had read Robert M. Pirsig’s *Zen and the Art of Motorcycle Maintenance* and decided that he would find the answers to the ultimate questions in the region between mathematics and philosophy. What he found, of course, was a very substantial dose of logic!

Sometimes it just seems that math can be a religion of sorts for a certain class of people, but as you have mentioned, we don’t deal with faith, we have concrete facts. For example many people say that the digits of $latex\pi $ execute a kind of mystical aura, but for me personally its just something which is part of the exquisite beauty of the world of mathematics.

What you have said about Buddhism is also /even more true for Hinduism if one can consider it as a religion. For example in Bhagavad gita, Krihsna after telling him about his views on Dharma and Karma, finishes with the statement

that you should carefully and critically think about what I told you and do what seems right to you. Similar is the views that one can see in Upanishads and the like texts of Hinduism. In fact if you look at the development of Indian philosophy itself, you can see that this view is present in Vedic period itself as nyaya (logic), sankya (observation, enumeration and deduction), mimamsa (investigation) tries to analyse critically the vedic/later knowledge in different ways. Unlike in most other religions, these were accepted to be part of Hinduism itself.

In fact in Buddhism after the time of Buddha, there is not as much freedom as in Hinduism to decide for yourself what is right and what is wrong.

Of course Hinduism is not a religion in the sense of western religions. In fact most of Indian origin religions are not comparable to western religions.

Yes, I completely agree. The debate about science and religion is in my view badly distorted because people take “religion” to mean one of the “religions of the book” and fail to notice that there are more open and accepting ways to relate to the transcendent.

If you put anything on the web, you expect that it will be copied. I already told the story of Neill’s illustration for the story about R. C. Bose working in the fields.

You might like to take a look here:

http://mum-anu.in/my-blog-2/

You will find this post, with minor modification (“my daughter” has become “my little cousin”), even using my banner and claiming to have taught the Oxford student referred to in the last paragraph.

Religion is based on faith, correct? So what is an axiom if not an article of faith? It cannot be proven, but is instead accepted as true and used as a tool for proof of theorems, etc.

#3 below is the key one here, since without it one can of course prove that “1” is equal to “0” (and by induction that all numbers are = “0”).

E.G.

1. “There is a number “0” such that 0*a = 0 for all a an element of the Reals”

2. “There is a number “1” such that 1*a = a for all a an element of the Reals”

3. “0 and 1 are not equal”

I’m afraid I disagree. I have many reasons, which I might write up at some point; here are two.

First, if (say) the axioms of group theory are articles of faith, and so group theory a kind of religion, then semigroup theorists (who deny the inverse law) and quasigroup theorists (who deny the associative law) would be regarded as heretics. On the evidence of the reaction of the Catholic church to Arians, Pelagians, Cathars and Protestants (among others), the group theorists would use all possible methods (not excluding war and torture) to suppress them. When I think of myself, Joao Araujo and Michael Kinyon discussing mathematics together, it is clear that nothing is further from the truth.

Second, if the axioms for a group were articles of faith, surely the person who brought the stone tablets down from the mountain would be revered? This is what Alexander Masters thought, when he wrote his biography of Simon Norton, “The Genius in my Basement”. He was amazed and disappointed that Simon didn’t know when and by whom the axioms for a group were first written down. But that, as Norton and every group theorist knows, is not how you do group theory!

I am not certain I understand enough about religion OR group theory to fully understand your answer. But I will take it on faith that you are correct 🙂

could one not say that due to the use of zero in mathematics that there in inherently faith as zero does not actually exist in our real world. In our world we have Infinitesimals but no zero its use in mathematics makes maths a faith.

Zero doesn’t exist in the real world? How much did you pay me to approve your comment then?

Seriously, though, I don’t need to believe in zero in order to reason about it. Euclid proved that there are infinitely many primes by supposing that there are only finitely many and reasoning about the finite list of primes.

And furthermore, saying that there exist infinitesimals in the real world is a much more drastic act of faith. Many cosmologists now think that the universe is discrete!

Euclid’s proof is subtly flawed, as there are not infinitely many primes. In fact all the evidence points to the inescapable conclusion that there are not infinitely many anythings.

Mathematics is a (written) language because it communicates (mathematical and logical) objects in our minds using (mostly) written symbols like ’0′, ’2′, ‘Pi’, ‘dy/dx’, ‘Fermat’s theorem’, etc. The ‘magic’ of mathematics is that every mathematics object, e.g. the integer symbolized ’2′, is the same though the object lies in different (people’s) minds. Even any mathematics object has been in our minds since we were born; children can distinguish different integers (one cake or two cakes for them before they can speak normally. So IMHO, the mathematical objects are (the only) universal things in our universe.

In some respects, Buddhism is in accordance with mathematics or physics as it considers its karma law as the extension of action-reaction law in physics to become universal (physics and mental) law. Buddhism also teaches people that there are many universes, and IMO, one of them can be a realization or a representation of an abstract space known in mathematics or can be the lowest (animal) universe. This can explain why some people, who had been in a more complex or more abstract universe, are ‘smarter’ in math than other people who had never been in that universe.

Unlike Hindu’s reincarnation, however, Buddhism teaches that the only reason we will be back (rebirth) in one of the universes is due to our attachments with worldly things. The combination of the karma law and the rebirth law is the only satisfactory explanation why some people (who made too many ‘deadly sins’ in their previous lives) have poor life despite the fact that they work honestly and hard every day in their current life.

Sir, per your…

“Does mathematics combine the mystical and the rational, as no other subject can? Maybe.

“Or maybe not. I had a student in Oxford who read Mathematics and Philosophy because he had read Robert M. Pirsig’s Zen and the Art of Motorcycle Maintenance and decided that he would find the answers to the ultimate questions in the region between mathematics and philosophy. What he found, of course, was a very substantial dose of logic!”

I hope that you do not mind this question…

Draw a horizontal line from zero towards infinity. Then draw lines from numbers below one to their reciprocals above one. Obviously, these lines connecting reciprocals cannot be distinguished from either the original positive number line nor the connecting lines. But those connecting lines are nonetheless there, yes? If so, without losing the reciprocal connections, to see those connections pivot a last portion of the original line vertically up off of the horizontal such that a first portion is going from zero to one is still horizontal, while a last portion is now vertically going from one towards infinity. If you do so, is there be any connecting line vertically up from zero or vertically down from infinity? I do not think so, and I am hoping you confirm that. I am also hoping you can tell me what area of mathematics might pertain to my observations that there is no connecting line whose diagonal becomes vertical no matter how close to infinity someone draws a connection from a point closer and closer to zero and to its reciprocal; plus this second observation: the distance of the top of a connector at infinity to non-connector line vertically above zero is dependent on the place of the pivot.

Warmest regards

P.S. Per the above unconventional metaphor for the positive real numbers, here might be where something religious for some mathematicians interested in religion might connect with something mathematical: if you approach 1 horizontally through the proper fractions from 0 at 1, you reach 1 as wholeness. Whereas, if you go vertically down from infinity through decreasing multiplicities, you reach 1 as oneness. Thus, wholeness and oneness are both the same and different in some kind of mathematical way, yes? As is 1 as unity, the union of wholeness and oneness. Sir, while I doubt any of this is of any great mathematical significance, I think it a way of using mathematics to highlight terms found in religion which are rarely seen as also mathematical. But I may be all wet, which is why I am reaching out to you for you unfettered response.

Interesting question: “But those connecting lines are nonetheless there, yes?”

When you ask if the connecting lines are “there”, are you talking about lines drawn with a tool of some sort, or idealized pure math lines that are 1 point wide? If the former then there are some purely mechanical difficulties in completing your cover of the positive half of the real line. If the latter then in what way are they there? I think we would need to set up some kind existence definitions in order to be sure we are talking about the same thing…

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I am one of those minority mathematicians that believes that infinity in mathematics is very much like religion. There are lots of things we would like to be true, and we use infinity to conjure those into existence. I have a blog entry called Infinity: Religion for mathematicians, at my blog njwildberger.com that discusses that a bit.

I am a mathematician, and I am very religious. However, I grew up thinking that math and religion were on different sides of an invisible fence. Over time I have learned this isn’t true. Math and religion teach us about each other. I manage a blog about this called “Theometry – The Math of God” at https://mormonsandmath.blogspot.com/?m=1 that you all should check out.

I agree that Math (and by extension any science) and Religion are deeply linked too. Isn’t this set of statements:

There exists a number 1 such that 1 x a = a for all a is an element of the Complex numbers

There exists a number 0 such that 0 x a = 0 for all a is an element of the Complex numbers

1 != 0

– based on faith alone? If we firmly believe that these things are true then we get answers that comfort us, but if we do not then all numbers are equal. So it works very well but there is no way of actually proving these statements – they are axioms and so can only be used to prove other statements.

And an axiom is a basic tenet of faith, no?

My view, for what it’s worth, is that there are no invisible walls out there, these exist only in our own minds. And on the subject of axioms, as I tried to say, an axiom isn’t an act of faith, rather it is an act of creation: it brings into being a new world in which we can play.

I believe you are referring to definitions, not axioms.

Still he’s right about using them to create things with. Well put.

I totally agree now that there are no invisible walls out there other than those that we create in our minds. I have now come to understand that every area of academia can give us valuable insights into every other area. About axioms, I agree they are an act of creation. That every axiom needs to be created suggests there must be a creator. Since the mathematical principles by which our universe is governed are universal, all things must have had the same creator. Consequently, I find that, even setting aside the power of spiritual experiences, believing in God is the most reasonable option.

And there’s always Pascal’s wager too…

Let me come back to Kacey Leavitt. I would be reluctant to attribute axioms to God. I could invent a silly concept and call it, say, a “quasi-hypergroup”, and write down some silly axioms for it. I might even persuade a few people to think about it and publish some papers, so that it might get an MSC number at some point down the line. But I don’t think I would have done anything like the axiomatisation of group theory in the 19th century.

There are two differences. First, group theory was already a well developed and important subject before the axioms were written down. Lagrange, Galois, Jordan, Cauchy had all worked on group theory. The axioms simply allow us to talk about a “group” in the abstract, not just a transformation group (say).

The other thing is that a robust concept like a group or a matroid can be axiomatised in many different ways. I wrote several posts on this a year or two back. An independence algebra of rank 1 is the same thing as a group acting on a set, although the axioms look completely different. Similarly, a matroid is a structure whose independent sets satisfy the exchange property, or a structure in which the greedy algorithm always works, or the dual of an IBIS family of sets (a definition modelled on computational group theory). I might agree if you told me that the concepts are created by God, but I think the axioms are our attempts to create something reflecting these concepts as accurately as possible.

Well said sir!

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I know nothing about maths. Seriously. Nothing. But I was wondering – If religious people create their God of the gaps’ (and they do) in order to have their cake and eat it would you say, mathematicians, so something similar by using ‘0’ and/or ‘infinite’ this and that? Are some mathematical theories dependent on the ‘finite’ and would break down without it?

I could say a lot about this, but instead I will be brief; I might return to this later, but today is a busy day for me.

On the face of it, there is nothing absolute about mathematical theories; they are created by mathematicians in a similar way to the way that God is created by religious people. But the astonishing things, which any philosophy of maths has to address, are: firstly, mathematics is consistent and stable over long periods of time; and secondly, our inventions are “unreasonably effective” in describing the real world. For example, Newton and Leibniz invented calculus, which relies on infinitesimally small quantities, which clearly don’t exist in the real world; and yet calculus underpins a huge amount of modern life via its applications in science and technology. It is not inconceivable to me that the same might be true in the religious context; let me just say, I have yet to be convinced of this.