Mathematics - Constructed By Humans Or Encoded In Nature
I was looking through Dayal’s lecture notes for the History of Astronomy (Part II), and decided I’d post some of my thoughts r.e. the questions on slide 40. "Is mathematics, the key to understanding nature, an invention of our human intellect? Or does mathematics have an objective existence encoded in nature, which we discover as we learn more about nature, and then refine this knowledge into mathematics?”

As someone who enjoys mathematics (excluding maths exams), these are questions that I find deeply interesting. There are several aspects of maths that particularly appeal to me:

1. the structural elegance and clarity of mathematical knowledge
2. the consistent reliability of mathematical statements (for example 1+1=2, regardless of the day of the week)
3. the strong links that are observed between abstract mathematical concepts and observations of physical reality

This last point, in particular, is one that I find especially intriguing. Why is it that computations involving abstract mathematical constructs (e.g. complex numbers) often give rise to valid solutions to real-world problems?

As to the question of whether maths is a purely abstract construct, or whether it has an objective existence encoded in nature, my opinion lies somewhere in the middle. I am inclined to think that maths is predominantly a human construct, in that it is a language borne of logic and therefore cannot exist independently of human thought. It is difficult to imagine, for instance, a collection of boulders engaging in meaningful mathematical reflection. (I realise this is a silly example, but it aims to illustrate my view that maths does not have an “objective existence encoded into nature”; thinking beings are required for mathematical thought.)

Why then, is there such good agreement between mathematics and physical observation? I feel that this occurs because of the way we define the foundational axioms on which branches of mathematics are built. The concept of ‘one’, for example, exists in an abstract sense. It also, however, has a firm basis in physical reality. Given that axioms are so often defined in relation to aspects of physical reality, perhaps it is not surprising that building logically on these axioms leads to results consistent with real-life observation.

But returning to the original question, perhaps something to consider further is that we ourselves are part of the fabric of nature. In this sense, both statements may be somewhat valid. Mathematics may indeed be an invention of the human intellect, but perhaps also, it is through human intellect, that the mathematical formulation of the universe has been encoded into the fabric of nature.

These are just some thoughts. I’d be keen to hear what others think about this question too :)

Josephine Davies (u5375415).

PS: I watched this video after going to Dayal’s History of Astronomy lectures. It contains some pretty awful attempts at humour as well as some equally awful visuals (you’ll see what I mean if you watch it), but despite this I thought it was a decent summary of the models that were discussed. But I strongly recommend that you only watch it if you think you can handle high levels of awkward acting and terrible jokes. You have been warned.

Very good! This topic is an old chestnut which has always divided philosophers and still does today. You'd think, as a philosopher of maths, that I'd be able to find something to disagree with, but I can't. I agree with all of that. (Well, except for the idea that only humans do maths — I'm sure alpacas do too, and later in the course we'll talk about aliens.)

Would anyone else like to disagree with Josephine? I would be amazed if we all agreed about this.


Interesting thoughts.
I have a different view, particularly in relation to something Josephine said about mathematics:

‘it is a language borne of logic and therefore cannot exist independently of human thought’

I think that’s an interesting view to take. I guess it also depends on how you define mathematics – whether you see mathematics as a formal language of symbols and logic (the actual process of writing out proofs/solving equations) or as something present in the universe (the actual relations or way in which objects interact with each other in the physical world). My view is the latter.

I think we should view mathematics as we do any other science. For example, does the theory of gravity exist independently of human thought? Yes, because even if we weren’t here, the force of gravity would still hold planets in orbit and cause bodies to attract each other. However, it is human thought that named this force, ‘gravity’, and which described the theory in words.

Similarly, even in the absence of human thought, things like 1+1=2 still hold. For example, a boulder and another boulder would still make two boulders even if humans weren’t there to add them. Or 3 rows and 2 columns of boulders would always produce the same number of boulders even if there were no words to describe the number, 6 (or 3 or 2).

My view is that mathematics is not defined as the formal language and proofs, but what this language predicates. The language that humans use to solve equations and work out proofs is a system that we have invented in order to make sense of mathematics (in the same way that we may speak English to describe the world around us).

Mandy Chau


I would disagree with the fact that a definition of maths is EITHER a formal language OR something present in the universe. Rather, they both work together - the language has been created in attempt to interpret and analyse events in the physical world. I would suggest (and hence somewhat agree with Josephine's last point) that neither can meaningfully exist without the other - we need a description of 'physical world mathematics' to make it a workable concept just as we need 'physical world mathematics' to create a language to describe it.

For me - the real debate is not which one exists, but whether the formal language we have created is an adequate description of the physical world.

In addition, I once heard an argument that there can NEVER be an adequate mathematical language because the universe is constantly changing - therefore, we would need a constantly evolving system of logic, not the static language of symbols and logic which we have… although not sure how I feel about this argument, yet to be fully persuaded.

Grace :)

Josephine raises some interesting points but I take issue with the following paragraph:

"As to the question of whether maths is a purely abstract construct, or whether it has an objective existence encoded in nature, my opinion lies somewhere in the middle. I am inclined to think that maths is predominantly a human construct, in that it is a language borne of logic and therefore cannot exist independently of human thought. It is difficult to imagine, for instance, a collection of boulders engaging in meaningful mathematical reflection. (I realise this is a silly example, but it aims to illustrate my view that maths does not have an “objective existence encoded into nature”; thinking beings are required for mathematical thought.)

I think declaring mathematics to be a language born of logic which cannot be independent of human thought conflates the method which we use to describe the world, namely language, with the portion of reality we are describing, namely mathematics/the relationships between quantities. It is true that the language which describes mathematics cannot exist independently of human thought, but to suppose that in the absence of humanity 2 + 2 should cease to equal 4 seems absurd. To put it another way, in the absence of humanity the symbol "2" would cease to represent the quantity of 1 + 1, but that one thing plus another is greater than a single thing would still be true despite the absence of some consciousness to acknowledge that fact and assign it the symbol "2".

Jake Stone (u5128350)

It's really awesome to be able to read everyone's opinions on this! :)

From reading some of the responses, we seem to be approaching mathematics from different angles (no pun intended), and perhaps that's why we disagree slightly with one another.

From my perspective, mathematics cannot be separated from language - I see maths as a language based on axioms and concepts that have originated from thinking beings. Whilst maths seems to be a really, really good description of physical reality, I don't think that these two things are the same.

I agree with the idea that "one rock + one rock = two rocks" regardless of whether we formalise that via mathematical symbols. But I don't think that the mere existence of the rocks (i.e. their physical reality) can be equated with mathematics - from that perspective I disagree with Jake when he calls mathematics "the relationship between quantities". In contrast I would define mathematics as "a language describing the relationship between quantities", and the relationships themselves as being something inherent in physical reality.

Then again, maths seems to describe the physical world so's pretty mysterious. Maybe an interesting question to ask (which ties back to Grace's last paragraph) is whether there is only one such form of mathematics - could there be other forms of mathematics that also agree with physical reality?

Josephine Davies (u5375415).

I would have to say that as far as the idea of mathematics not being able to be separated from language, I am in complete agreement with Josephine. We cannot separate our understanding of the world around us from our thoughts and expressions of such, which is language. From a structuralist sociological perspective, it is impossible to accurately account for the world around us in an objective manner. We are born into our society with its language, thus our thoughts are a product of the language we are born into. We are limited in what we can think by what is around us at the time. It isn't for nothing that scientific 'discoveries' are made by multiple people around the same time in history - with the same, or very very similar information, the next stage that is accepted will inevitably be similar.

Where I would particularly like to prod at though, is the idea that no matter where you are 2+2=4. No it doesn't. There are tribes that count - 1, 2, many. That's it. Everything above 2 is many. For them 2+2=many with no differentiation in their language and hence a very different understanding of the difference between 3, 5 and 27 in the way that we do.

I would like to be clear here that I am not suggesting we abandon mathematics, but I am suggesting that it makes sense only to those it makes sense to. I will agree with anyone who answers 4 is the answer to 2+2 here in Canberra, or arguably anywhere that accepts the numerical system under which we operate. I will also agree with that answer if I'm surrounded by people who would answer many, and I will then have to suffer the consequences of the misunderstanding that stems from it.

If anyone is interested at looking at some other numerical systems, check out the following link:
Some pretty interesting ones in there.

Cat Gordon (u5024359)


I found this a very interesting discussion, which is strange since it is about maths which usually would make me tune out!
I think Josephine put it quite eloquently relating mathematics to a language of logic we use to understand the world. It becomes a way of categorizing things and predicting and quantifying phenomena. While the phenomena exists outside of the categories we impose on it, we can only explain and describe it using the language we have created to do so. Take Cats' example of variations in cultural concepts of numbers, people across these different cultures would theoretically have different answers to mathematical questions depending on the language they used to frame that question. The meaning of numbers is subjective based on the way the number is described.
But why does maths seem to work so well?? It must be real/describing the truth since it is so useful, right? I'm not a realist, but even I am struggling with the though that maths could be a human construct.
I think I need to do some thinking about this one and come back with some ideas :)

Kathryn Reid (u5194612)


As I see it, mathematics is a formalisation (not formal language, but formalised notions) of applying propositional deduction to our definitions of abstract entities. The definitions are motivated (or inspired) by what seems intuitive due to humans’ primary experience of non-abstract entities in the physical world. Therefore I don’t think mathematics has as ‘objective existence encoded in nature’, but it is a result of the fact that it is constructed by humans for the convenience of humans.

Also I disagree with Josephine’s argument that because humans are a part of nature, mathematics has been encoded into nature through human intellect. There’s no reason any two people’s definitions should be the same, although similar motivation may arise due to similar primary experience of the physical world. Since mathematics is just a set of intellectual notions, we could easily say that their own mathematics is valid based on their own definitions. However in the field of mathematics we choose specific notions to be universal. Hence there’s no reason why this chosen mathematics should be encoded in nature any more than any other.

I’d like to emphasise the importance of definitions in mathematics, agreeing with Kathryn’s statement ‘The meaning of numbers is subjective based on the way the number is described’. There isn’t any objective truth to ‘1+1=2’ unless we define 0, 1 and then 2 by induction, the operation of addition and the equivalence relation ‘=’ on the natural numbers. Furthermore, these definitions are certainly not encoded in nature. If they were, it would insist that we define the natural numbers as groups of objects with a certain property (3 apples, 10 pink Ferraris, etc), but we can’t do so because groups of objects also don’t objectively exist. The universe doesn’t classify, humans classify (because it is convenient). So we define the natural numbers using set theory: as unions of sets of nothingness.

I disagree with Cat and Josephine about not being able to separate language from mathematics. ‘Formal language and proofs’ is not mathematics, it’s just there for convenience. What people do when they’re ‘doing mathematics’ is really just investigating the consequences of definitions. The medium used is an arbitrary means of representing intellectual ideas (write it, speak it, paint it in a cave for all we care). We choose to have some conventions of notation for ease of communication. Neither is working out ‘the velocity of a particle at time t=pi/2’ actually mathematics either. It’s calculation, which is just approximating the parallels between the definitions in mathematics and the problem at hand, then using mathematics to suggest what an answer might be without actually measuring it.

An illustration of this principle is how we draw parallels from the construction of the natural numbers by unions and the union of objects we have classed together such as apples. By these parallels, we can use the properties of the natural numbers such as addition to approximate the answer to ‘what do we have if we have 3 apples and then another two?’. We approximate 3+2=5 by the inductive definition of the naturals, and can verify the accuracy of this approximation by counting the actual apples. So I agree with Kathryn saying ‘It becomes a way of categorizing things and predicting and quantifying phenomena.’

This also addresses Jack’s argument that ‘one thing plus another is greater than a single thing would still be true’. The physical truth of this is undeniable, but this is not a mathematical truth. For a simplistic example: one of the axioms of the integers is the existence of additive inverse, by which I mean a mathematical object (which we happen to call a negative number) such that addition of this object to its strictly natural counterpart is equivalent to zero. The set of integers doesn't use a notion of ‘taking away’ anything, so this is addition as valid as any physical truth, however it doesn't necessarily give the ‘greater than’ result which the physical truth insists. Hence, mathematics only shares this physical truth if we restrict the mathematics some way. The physical exists independently of humans, but the mathematics doesn’t.

That was lengthy, but overall it should become apparent that I think mathematics is just a human construct.

Christie-Rae Crosthwaite (u5366683)

Most of these topics are too hard for me!

But please keep up this discussion. There's lots of great stuff here.

A couple of things for Christie to think about:

1. If the ONLY definition of addition was on natural numbers, using von Neumann's definition or one of the other definitions in which natural numbers are just sets of sets (of nothing), then addition wouldn't (by definition) apply to physical objects. I think this is pretty uncontentious, even among set theorists. So we need either a different definition of addition, or multiple definitions. There are lots of ways out of this problem, and I'm not sure which is best.

2. It's not clear to me that addition actually DOES apply to physical objects, in some cases. In other cases it does. We actually need a theory about when it does and when it doesn't. This is not a new idea by any means (IIRC, Russell mentioned it a hundred years ago, and no doubt lots of people before him) but is a topic that often gets overlooked. For example, one cloud plus another cloud, if you put them close enough, do NOT make two clouds. But one apple plus another apple DO make two apples. In practice we don't have too much problem knowing when addition is going to work and when it isn't ... but if we think we're DEFINING addition then we need to be explicit about when it will work, and that is not so easy. It might be tempting to say "1 + 1 = 2, provided we're talking about normal objects and not things like clouds", or "1 + 1 = 2, provided we're talking about separate objects and not objects whammed up against each other", or "1 + 1 = 2, except when it doesn't", but I hope it's obvious that we need more detail than that, or we haven't tactually got an explicit definition.


1. I disagree a bit with you there. Where ‘nothing’ is nothing and is defined as zero, a ‘set of nothing’ is something. It is a set, so a mathematical object in its own right. So when we have our set, (which happens to be 1 once we’ve given the whole inductive definition) we can use it to represent our physical object. So although I agree with you that by definition, addition on the naturals does not apply to physical objects, however it does by definition apply to an abstract object. Hence calculating the addition of physical objects using mathematics, we’re saying ‘Suppose that our apple, wasn’t physical and that it was the abstract set {0} (ie. 1). Then the addition of 3 apples and 2 apples would be defined the same way addition on the naturals is defined, so we can compute the result on our abstract mathematics and assume the same for physical quantities’. So I don’t think we need a new definition of addition since mathematics isn’t necessarily for calculation purposes, we just need to remember that we are drawing parallels between physical and mathematical definitions and not actually using the physical to define the abstract or vice versa.

2. Given the above, I don’t think addition does apply to physical objects. So perhaps instead of defining addition, we ought to create rules about when physical objects are distinct from each other. For instance, it may be more helpful to consider all of space as the metric space R^3. Then the naturals are discreet in R^3 as they are a distinct subset of the reals. Then addition on the naturals the only applies to physical objects which we define to be distinct from each other. So in your example, we should define the clouds as distinct as long any two particles in different clouds have a distance greater than say a metre between them. If this is the case, we can use addition on the naturals because we have more than one object, otherwise they are the same object in our discrete space and we have no need to add anyway.


I found Christie’s points really interesting, and I agree with most of them. I definitely agree that there is a strong element of human construction in mathematics.

I just wanted to clarify what I meant when I called mathematics a ‘language’. As Christie points out, it is perhaps not accurate to equate mathematics with language (at least not language in a conventional sense). When I speak French, Arabic or Urdu (perhaps in some parallel universe where I tried a lot harder than in this one!), I agree that it’s not quite the same as doing maths.

Take English, for example. Although there are ‘definitions’ and ‘grammatical rules’ that I should know if I want people to understand me, these rules and definitions haven’t always been the same. Language evolves - vocabulary and grammar change with time, and there is no ‘right’ or ‘wrong’ way for language to develop. As long as people can understand you, all is well.

In maths however, I don’t think it’s quite so arbitrary. Whilst the axiomatic basis of mathematics may differ between cultures (Cat’s link to numerical systems has some great examples), the rules of logic we apply in maths seem to be quite universal.

So I guess what I actually mean when I called maths ‘language’, is that maths is a kind of formalised logic that shares elements of language; it has an ‘axiomatic vocabulary’ and a ‘logical grammar’. And whilst I agree that the axioms are somewhat arbitrary, I don’t think that the same can be said of the ‘grammar’ of mathematics. In the logical world, some statements make sense, and other statements don’t make sense. If I hate all boys, and my brother is a boy, then the statement “I love my brother” just doesn’t make sense.

But then, what does it mean to ‘make sense’? I guess that’s what I find interesting; what is ‘logic’ in the first place? Is logic itself, separate from any arbitrary axioms and definitions, a human construct?

Josephine Davies (u5375415).

Christie: I like the points you make. I think I've read what you wrote carefully, and I can't see why you say you don't agree with me! Anyway, to blather randomly on about the topic, there are versions of maths that try to apply addition to physical objects, although they're currently considered pretty non-standard. For an example, see (which is VERY long, but really good if you've got time to read it).

Jo: I'm a big fan of your contributions to this board (I hope you're going to do at least Honours in philosophy), which makes me feel free to reply to:

"the rules of logic we apply in maths seem to be quite universal"


Maybe one day we'll have universal rules of logic that we can apply in maths, but let me say a bit about where we're up to so far.

Logical systems detailed enough to have any hope of capturing what we do in maths didn't exist until 1910, and then many mathematicians jumped on the first one that came along (roughly - the history is fascinating, but this is not too much of a simplification). Logicians, on the other hand, have never agreed what the best system is, and have pointed out plenty of problems with the most standard system.

And it actually does make a big difference to maths.



Wow, this stuff is really interesting - I can't stop thinking about that 'Penguins rule the Universe' proof!

I guess I'll need to think about this question more carefully. Maybe a better way of thinking about it might be to think about mathematical structure. When Cat posted that link to all those numerical systems, it all seemed really arbitrary, i.e. I can choose how I define the number '1', and it might be different to your definition.

But I still can't help feeling that there is an underlying consistency in mathematics. After reading Jason's post I'm convinced that this consistency probably doesn't lie in logic, so perhaps it lies in the more general structure of mathematics.

For example, I can describe the relations between a particular set of abstract entities using various forms of mathematical notation (e.g. using matrices, words, algebraic equations, etc). But switching between these forms of notation still preserves the relations between the abstract entities - in this sense the mathematical structure of the relations is independent of notational definitions.

Whether mathematical structure is in itself a product of the human mind, I'm completely unsure. (By the way, given my limited math background, I'm probably saying lots of dodgy things, so apologies to any hardcore maths enthusiasts out there.)


There's a fashion at the moment for describing your view using category theory: I don't know much about it, but it seems like a good idea.

There's a related idea that maybe not only maths but the whole of science can be described using structures that are stable even through scientific revolutions. This is called "structural realism". I can't see how to make it work, but I think it's a very good avenue for research. See (Everything I cite here is examinable, by the way.) (Just kidding — it isn't.)


Jason: I disagreed about needing different or multiple definitions of addition because it would be more consistent for the rest of mathematics to redefine the distinction between objects instead.

Actually, consistency of previously proven results is one of the reasons working mathematicians predominantly use ZFC constructions today. I finally got around to reading that set of notes over Easter, and so I had to have a chat to my analysis extension lecturer about why he teaches us ZFC and his opinion on New Foundations + infinity +choice. I found it really interesting how NF allows a universal set and still avoids Russell’s paradox, whereas ZFC must disallow the universal set to avoid the paradox.

NF also seems to rely heavily on the quotient of a set by an equivalence relation . ZFC uses a the quotient by equivalence relation, but that’s mostly in the idea of ‘completing’ the rationals to form the reals and other sets which like the reals, have a notion of an infinitesimal distance between members of the set. NF however uses the quotient to define the cardinality of a set, which is an interesting choice since the idea of the quotient has its origin in geometric and topological constructions (eg. Define a circle by taking an interval [0,1] and define the equivalence class of [0] to be 0 and 1, meaning 0 and 1 are at the same place on the interval and thus defines a circle).

The notes you linked seems to have a more ‘natural’ definition of addition where it is essentially the cardinality of two sets, which is how we think of it when we count 3 objects in one set, 2 in another and conclude that the addition of the two sets is the addition of their cardinalities = 5. This is probably only more ‘natural’ when we’re talking about the addition of the natural, integer and rational numbers. Once we get to the real numbers, addition is pretty much the same in both NF and ZFC.

What I think is less ‘natural’ about NF in the notes, is how it defines the integers using the definition of the rationals. It’s more natural really to build from the ground up like in ZFC where one begins with two identity objects 0 and 1, to define the naturals, which then defines the integers, and rationals by their extra properties on top of those and then completing the rationals to form the reals.

Jo: I like the idea that the underlying consistency in mathematics may lie in the structure of mathematics, and I think it partly does, but only because of the historical timeline of mathematics. The foundational crisis which occurred around the beginning of the 20th century was well after a lot of more advanced theorems were proven assuming commonplace notions number, etc. It would have been unsatisfying to have the different branches of mathematics built on different foundations because some of the strongest results combine areas of mathematics such as analysis and algebra (maybe that’s why NF above decided to use a more topological idea like the quotient space to define a set-theory idea like cardinality). Most working mathematicians accepted ZFC because it seemed consistent among the branches and they wanted to get on with their work, but many set theorists dispute this which is why there are many alternative set theories. See
So assuming one logic and axiomatic set theory for all branches of mathematics, I think you’re correct.

Jason brought up constructivism. The various schools in the philosophy of mathematics are really interesting and find myself mostly sympathetic to logicism, and maybe (by a bit of a stretch) formalism, but I find constructivism and intuitionism hard to swallow. I think this is mostly because there are a lot of powerful existence theorems (including the axiom of choice) at very base levels of most fields which rely on a proof by contradiction by assuming the non-existence of an object and drawing a contradiction. This proof is disallowed by intuitionism and constructivism, so there are some mathematicians who are attempting to obtain these theorems by constructing an object, usually by employing much weaker consequential statements. This is why constructive mathematics is about 100 years behind the others.
Is there a course where we can study things like this more in depth? Because I'd really like to have discussions about this more.


Thanks. Very interesting!

You know, I think there IS a philosophy of mathematics course, but I've never had anything to do with it. Look on the web or ask one of the admin people.

Also I supervise philosophy of maths research projects sometimes. See for previous ones.

More later.



This topic has great discussion! The maths far exceeds my familiarity with the subject but the blurry outlines of the theory have still served to elucidate my thinking.

A common theme I noticed throughout was an appeal to intuition. Whether it was Jo saying she felt there was an underlying consistency to mathematics or Cristi saying that she liked an idea of an underlying consistency.

Being somewhat an epistemic nihilist (I know, I know, how tiring) I find appeals to intuition unconvincing. And this got me thinking about Carnap's theory of Logical Tolerance which posits that their is no single underlying logic and each individual adopts arbitrary axioms when constructing their system of logic. Given the definition of an axiom, that they cannot be justified seems to me an uncontentious point to make. So logicism, the idea that mathematics can, at least in part be reduced to logic, becomes a constructivist model.

Jake Stone

Certainly there is no single underlying logic ... or, if there is, nobody knows what it is or how to show objectively that it's the best.

But thinking that axioms are arbitrary goes way way beyond that. (Did Carnap really go THAT far? I don't think so ... although I could be wrong, especially since he changed his mind about various things related to this.)

I'm tempted by the idea, but it doesn't seem to work very well. For example, the following axioms are not very useful:

1. • implies §

2. § implies •

3. Any number of horses may gather between 6 and 9 pm.

I don't know WHY that system is not as useful as mathematical systems, but I'm pretty sure it isn't. Nor would anybody count it as a logic or a mathematics. So what IS a logic or a mathematics? I wish I knew.



If something can't be justified how can it not be arbitrary? You pick it based on a personal preference. Which is what you did Jason in constructing your counter argument above.

You decided that usefulness was the litmus test of a logic system, presented a system that was not useful and so dismissed it. The arbitrary step being that usefulness should take precedence.

The system may be coherent after that initial decision to place a preference on utility, but the foundations remain unjustified, except for an appeal to the vagueness of intuition, which I have already dismissed (though I find when I argue about this I often dismiss intuition to my own satisfaction but fail to convince others).

Jake Stone

Right! If something CAN'T be justified, it's arbitrary. So either it IS arbitrary, which I admit is possible but which I find implausible, or it can be justified. Just not by me. Because I has teh dum. The fact that I can't do something is really no proof that it can't be done. (This applies to changing the oil in a car as well.)

What we mean by usefulness is a tricky topic which we should come back to some time.

By the way, in other areas I think philosophical intuition is vastly over-rated (and I think this is a serious problem in contemporary philosophy). We could talk about other examples, especially in metaphysics. I'm just not so sure about logic and maths.


Just bumped into this video: Is the Universe Entirely Mathematical? Feat. Max Tegmark.

It's definitely very relevant to the discussion we've been having!