Many people dislike math. They think it intimidating, cryptic, and often divorced from reality. Yet math in general is practical, accessible and so easy, that we did it long before we heard of math. In fact, math may simply be an abstraction of the most fundamental aspects of our psychology.

**Warning: Bad Pun Ahead**

For instance: why do numbers count?

Maybe numbers don’t matter, but their relationships do? Maybe numbers are just place-holders in the relationship? Let’s take a concrete example.

When I think of things, I often try to order them in some way. I think of one thing as better than another, some tree as higher than another, some food as tastier than another. Now the goodness of a thing, the height of a tree, the tastiness of a food are all different things, but the RELATIONS among these respective groups are the same, and it is this that I try to capture with a number.

So I abstract the relation from the above, and introduce numbers as a place-holder in this relation. Abstraction means many things, but in this sense, I’m using it to mean a general principle underlying multiple concrete ones. In the case above, this abstraction is GREATER THAN or >. I would then record this general principle (abstraction) as:

*x > y*

The generic single letters (x,y) are no accident. I’m specifically being “ambiguous” since I could be talking about trees or food or people. By keeping neutral (even cryptic) lettering, I fight against the tendency of some to reify this general principle to concrete cases. Unless I want to calculate something specific, I want to focus on the RELATION itself and not what it is in specific instances.

Most importantly is the shift in thinking: x and y are place-holders and the real star is the relation itself. So when thinking numbers, perhaps my shift should be the operations themselves?

This is why math can seem so cryptic. It’s abstract, then mathematicians find abstractions governing those abstractions and so on. At every stage, an attempt is made to preserve generality and so those looking for something concrete to grasp may feel frustrated.

**Would You Eat This?**

I try to group similar things. For instance, if I taste chocolate, feta cheese, 3 day old bread, and wet toilet paper, I’m likely to categorize the first two the same way: things that taste good.

I just formed a set (there are more restrictions, but this is good enough for now). Now we come to Closure. This is a VERY intuitive idea that just seems cryptic because of how it’s presented. For instance, a common definition would go something like this:

*The set X is closed under f if for all a,b in X, f(a,b) in X.*

Sounds cryptic? It ain’t. You use it every day. In fact, you use it when you cook. We already know what a set is, f is some operation that takes two things in the set, and a,b in X simply means a and b are any two things in the category (set) of X. So what we’re saying in plain English is:

*We consider a category closed with respect to a certain operation if using that operation on things in that category ALWAYS produces a thing in that category.*

It’s a bit more readable, but maybe a concrete example will drive the point home.

I have a set of tasty foods, which include things like chocolate, marshmallows and feta cheese, Now I want to start mixing them — that’s my operation. Well, if I mix any two tasty foods, is the result ALWAYS tasty? Well, chocolate and marshmallows taste good (think s’mores), but chocolate and feta cheese do not. So the set of tasty foods is not closed under mixing!

Closure is a great thing because it allows us to chain an operation. For instance, I presented the operation as only operating on two things, but in truth it can operate on any number of things because of closure. Remember f(a,b) takes any two things in X and produces a result in X. So this means I can take the result of f(a,b) and apply it to f again! For instance, if I wanted to do the operation on 3 things in X (a,b,c), I could do:

*f(a, f(b,c))*

Since f(b,c) is in X. And I can repeat the process indefinitely, I couldn’t do this if f wasn’t closed, because one of the results would not be in X and the whole thing would become illegal. This is one reason why I can chain operations like addition. Addition can be treated as a function with infix notation. That is, rather than +(a,b), I can just do a + b. So the chaining of a + b + c is the same as a + (b+c) which is the same as +(a, +(b,c)). All possible, because addition is closed.

In fact, software developers can recognize this as a potentially useful property, all too often because the code they work on lacks this.

Which brings us to another point: sometimes consequences arise from these properties that aren’t obvious at first glance (or are completely surprising). In fact, much of mathematics was built on tracing the consequences of these ideas, and this underlies proofs and axiomatic systems.

**The Mathematics of a Broken Heart**

Take commutitivity. It just sees if the order of the operation matters, or in other words:

*f(a,b) = f(b,a)*

To see its applicability to daily life, we can define unrequited love in terms of the above. Let f = love, let a = John and b = Mary. Saying John’s love of Mary was unrequited is simply saying:

love(John,Mary) != love(Mary,John)

Unrequited love is non-commutative love!

There; mathematics just saved your relationship. You can thank me later.

**Conclusion**

I’m not “dumbing down” mathematics; mathematics “smartened up” the things we normally do. Since math is an abstraction of the way we see the world, it studies us.

Or does it?

How logical are we? If I take an abstraction of several ideas, and trace their consequences very rigorously (eg: via proof), then have I arrived at something that bears any relationship to human thought? Does the mind operate that way?

Have I moved beyond the thought to studying the ideas themselves?

Which brings us back to the question:

Is Math a psychology?

Or is it just a study of ideas?

I’ve been thinking about this for a while and I’m not sure I have anything useful to say. I’ve studied maths at university level so I’m familiar with mathematical thought processes. However, I have to be completely honest and say that I’m not entirely sure I understand your questions. Perhaps I have my feet too firmly in the maths camp to see the bigger picture. My husband is a philosopher so I am somewhat familiar with that camp also; enough to know what I don’t know! I think that the formal logic that my husband studies is closer to a psychology than any branch of maths that I’ve studied. However, my experience is that humans are, on the whole, not logical. 😀

Actually, what you wrote was very useful. At the very least, it showed that I should clarify.

Many fundamental mathematical structures appear to be abstractions of human ways of perceiving the world. So would math qualify as a psychology? If not, why not?

My question may be a regurgitation of psychologism.

One objection to claiming math is a psychology is that these “perceptual structures” are so abstracted, they no longer reflect the way human minds work.

Your reference to formal logic is very relevant. Some took logical studies as an attempt at figuring out how the mind worked, and while it might have captured some things, I think it ultimately failed to provide a model of how we think. But it was useful in its own right 🙂

In fact, wasn’t Principia Mathematica an attempt at founding math on symbolic logic? If so, then the connection to human thought becomes more pronounced.

By the way, my mathematical knowledge is limited as my only formal training was minoring in it many moons ago. Everything else was self-study.

I think I would agree with the idea that maths does not reflect the way we think. Unless you are able to isolate and abstract ALL of the inputs to a particular thought process then you aren’t going to be able to model it successfully. Since most of the inputs to our thought processes are subconscious then that is a tall order. I think that’s where fuzzy logic comes in but I don’t know much about that.

As far as I know, Principia Mathematica was indeed a monumental attempt to construct an internally consistent mathematics based on symbolic logic. We have a copy (3 weighty tomes) that my husband is working his way through – it is mind-boggling. When I come across works like that I just wish that humans lived longer so that great minds like Russell and Whitehead could follow their ideas through to completion. By the way, my husband tells me that PM was not broken by Godel (sorry, I don’t know how to do the umlaut mark) as many people believe.

Excellent point. If math/logic models anything, it’s how we think we think about high level concepts, which doesn’t take many factors into account, including subconscious inputs. A true model would be lower level, and fuzzy logic/neural networks look promising.

I actually peeked at PM and backed away because I didn’t want to learn a different notation for logic/set theory. I wonder if there’s a version that’s been updated to use modern notation?

I assume you’re referring to Godel’s Incompleteness theorem? I never thought of its applicability to PM as the challenge I’ve always seen raised to PM was Russell’s Paradox, which he resolved with types. Since PM is “just” an attempt to provide a foundation to math, would any issue with math’s inconsistency even be relevant?

I hear you on Russell and Whitehead. Great thinkers in their own right, and that’s before they successfully turned their hands to philosophy.

Russell has influenced quite a bit of content in these articles. His views in “Logic and Mysticism” shaped my attitude towards math, and I quoted whole swaths of the last chapter of “The Problems of Philosophy”. I need to (re)read more Russell.

Speaking of people I need to read more of, I do want to read more Whitehead. My main familiarity with him is via Process Philosophy.

So many books, so little time.

“So many books, so little time.” That’s been heard a few times in our household too!

I’m not sure I know how to answer your questions. I’m hoping they were rhetorical ;).

Lol! They’re both rhetorical and non-rhetorical. How’s that for violating the laws of logic? :D.

That was rhetorical, heh.

😀

Sounds like a quantum superposition to me…

ROFL!

Hi! Took me a while to read this, again, life is hectic…

First, I have to say this is one of the most fun posts I’ve read about math. Teaching this myself, I struggle to get my students to realize they already know how to do most of the stuff. For example, most students will memorize distance = velocity x time, but none of them will think about this when they look at their car’s speed-meter and guess that, at 100 km/h, in 3 hours they’ll have gone 300 km. This completely confuses me. How can they know how to do it and yet not know how to do it?

I also remember having this student who couldn’t tell me how much 20 x 5 was. So I asked her: if you have 5 20-peseta coins, how many pesetas do you have? To which she replied: 100. Then I said: great! So how much is 5 x 20? Answer: I don’t know.

It’s like they believe math and physics to operate at this level that has nothing to do with reality whatsoever.

Mind-boggling.

That said and regarding your question, I’d distinguish between two different questions:

1. Where does math come from?

2. What is math?

The answer to the first question may very well be a psychology. I believe there are many answers to the second question. For example, you could say math is the set of all axiomatic systems and their theorems (taking logic to be another axiomatic system). If you take that approach, you could then make the connection: axioms –> instructions and say that math is the set of all possible computer programs. You could view math as a completely formal system or you could take the intuitionist approach. My personal feeling is that all of those interpretations are just facets of the same thing, but I could be wrong.

Anyway, those were my fifty-cents. Off to read the other post I missed…

Thank you!

I’m glad you found this post fun, because that’s one of the points I try to get across with these math articles. Math is fun, friendly and anyone can do it. Sure, not everyone can work with some of the advanced topics, but there are tons of accessible, fascinating math subjects.

Good point on people’s ability to solve the same problem in a different context. Is this a failure of abstraction? They know the rules, but can’t take that next step and realize that a number need not be connected to anything concrete (like distance).

I like your breakdown of the question into 2 different questions, and love your answer to the second, especially since I’m on an axiomatic kick and am fascinated by axiomatic systems. In fact, I’m reading a book right now that covers axiomatic systems like Euclid, Hilbert and Peano.

The tie between axioms and programs is an interesting one, and it really shines in the cases of functional programming and term rewriting systems. With imperative programs, there’s a bit of a gotcha in the nature of an imperative…

I always thought the intuitionists were fine with formal/axiomatic systems, but just had issues with certain concepts like infinity and the law of the excluded middle?

What an interesting discussion. By coincidence I’ve just finished a post on almost exactly this topic. I’ll be saying that Russell did not ‘successfully turn his hand to philosophy’, that his symbolic logic is not useful in philosophy. I believe that his Principia project failed because he made a simple error closely connected to his psychology and thought-processes, and that his metaphysics failed for the same reason. For a successful mathematical model of how we think that does not run into Russell’s paradox there is his colleague George Spencer Brown’s ‘calculus of indications’. This is the model that accords with Kant and Jung’s psychology. It’s very well worth checking out his book ‘Laws of Form’. It reaches the parts that Russell cannot reach.

I love the discussions that ensue in comments.

I’m checking our your blog now. Metaphysics, consciousness, math… very nice. Would you mind providing a link to that article that was almost on the same topic?

I’m also going to check out Brown’s book and read up on the Calculus of indications.

What are your thoughts on Brouwer and his intuitionism? He also wrote an interesting work “Life, Art and Mysticism” that I’m still working through, but that you can find here:

http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.ndjfl/1039886518

It’s unclear if the link above is an abridged version or not.

That was an interesting read and really broke down maths into an easy to understand model. As far as i can understand it, it seems that thought processes are just extremely complex logical algorithms, so complex in fact that to us it can seem random, and if you ask me the same goes for all existence, the combination of different atoms creates different molecules, and the placement of each molecule creates a different effect times that one million trillion billion and you have a world that seems to be working at random.

Maths is just the part of that complex programming and engineering that we can understand, maths isn’t a part of philosophy, its that where maths stops philosophy carries on. They are the same thing in which us humans have separated and classified into what we do understand and what we don’t.

I have also always been interested in the maths of music and emotions, how the relationship between different frequencies and the length they are played at can have a resultant effect on the emotions of a person, a good example of the relationship of maths and philosophy maybe?, if there is anything you know on this subject i would like to hear it

It’s interesting you mention physics because there are some out there who claim that the universe IS computation.

Regarding math and music, there are some connections, and I think the Pythagoreans explored some of them, particularly the ratios involved in pleasing frequencies. There has also been work on automatically generating music via a few different methods, such as rewriting systems and Fractals. I’ve actually done the latter two and have produced some pleasing tunes, that I sadly no longer have.

I’d be happy to provide links for you if you wish; are there any particular subtopics of music/math from what I mentioned above that particularly interest you?

The relationship between musical frequencies, tonalities, dynamics and so forth, and how they affect us, is a fascinating subject. I think Hans Keller has most of the answer with his ideas about the manipulation of our expectations. But why do we expect anything in the first place? This seems to be a matter of maths and partly to do with cultural conditioning, and I’m not sure ti would be possible to disentangle them.

David – What you say about your students is very interesting. It made me laugh. I’d never thought of this before, that people can often do the sums when they need to but not when they are given as abstract problem in maths. Hooray. Maybe people who are bad at abstract maths are not always as stupid as they seem.

It made me laugh too. I think the basic problem here is some people think that what they learn at school bears absolutely no relationship to anything in their real life. Therefore, they can calculate how long it will take them to get to Paris if they drive at 120 km/h but they can’t do kinematics; they can count money and perform complex operations with it but they can’t do math. And, in my experience, it is really hard to make them see the relationship.

Yes, I remember an author trying to analyze music in terms of the psychology of expectation. Why would we expect anything? A good, fundamental question that can affect our lives 🙂