Was Mathematics Invented or Discovered?

Date: 01/01/2001 at 15:37:07
From: Angela
Subject: Was mathematics invented or discovered?

Dr. Math,

I'm currently taking a course in high school that studies (along with 
philosophy and logic) mathematics. We have been assigned to write a 
1,000-word essay answering the question "was mathematics invented or 
discovered?" Can you please help me with this difficult question? From 
my point of view, it can be argued either way. I realize that you 
addressed this question back in 1995, but I think it is a question 
that deserves to be elaborated on. Or, if not, could you direct me to 
some Web sites that will give me different viewpoints as to the 
history of mathematics?

I personally think that the concept of mathematics was discovered. You 
can't "invent" 1 + 1 = 2; it just does. I'm confused as how to present 
my viewpoint. What other possible arguments are there? It will be very 
helpful if you could give me ideas from both sides. Thank you very 
much for your help. It will be greatly appreciated.

Angela

Date: 01/02/2001 at 13:59:32
From: Doctor Ian
Subject: Re: Was mathematics invented or discovered?

Hi Angela, 

Let's begin with your assertion that "You can't 'invent' 1 + 1 = 2; it 
just does."

On the contrary, 

     1 + 1 = 0                       in mod(2) arithmetic

     1 + 1 = 10                      in base(2) arithmetic

     1 + 1 = 1                       in Boolean arithmetic

     1 + 1 = [an arbitrary value]    in group theory, depending on 
                                     your choice of group.

In fact, '1 + 1 = 2' is a consequence of selecting one particular set 
of rules to define one particular formal system. But there are lots of 
other systems that can be defined, and confusing 'Peano arithmetic' 
with 'mathematics' is like confusing 'Chess' with 'games'.  

You should probably take a look at the following discussion from the 
Dr. Math archives:

   Understanding Mathematics
   http://mathforum.org/dr.math/problems/nwasokwa.7.25.99.html   

But you should also give some thought to whether 'invented' vs. 
'discovered' is a distinction without a difference. Does the answer to 
any other question depend on the answer to this one? Would it affect 
how mathematics is created, or used? Would there be any discernible 
difference between a world in which mathematics is invented and one in 
which it is discovered?

To get a little practice thinking about this kind of thing, consider 
the question:  do human beings have free will? Is there any outcome or 
situation that can be explained by free will that can't also be 
explained by predestination, and vice versa? There isn't. In which 
case, what is the point of asking the question?

(If you don't understand what I'm getting at, find a copy of the book 
_Pragmatism_ by William James and read it. In fact, you should read it 
anyway. It will save you a lot of useless head-scratching over the 
course of your life.)

Personally, I think that the question 'Is math invented or 
discovered?' is meaningless. But thinking about _why_ it's meaningless 
can lead you to some very interesting ideas about the nature of 
meaning! (For example, if you're lucky it might lead you to read Doug 
Hofstadter's wonderful essay 'A Conversation with Einstein's Brain', 
in his book _The Mind's I_.) So I hope you'll continue to noodle 
around with this issue this even after your essay has been completed.

I hope this helps. Feel free to write back if you'd like to talk about 
this some more, or if you have any other questions.

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/   

Date: 01/03/2001 at 17:26:17
From: Angela
Subject: Re: Was mathematics invented or discovered?

Hi Dr. Ian:

Thank you so much for your quick response.

Although I agree with your response that "the question 'Is math 
invented or discovered?' is meaningless" I have no choice! I have to 
write the essay anyway. And it will be difficult to come up with 1,000 
words on this topic. I read the discussion that you suggested, but I 
am still confused. So, I am writing back to ask you some more 
questions. Here goes!
 
1) What exactly is Peano mathematics?

2) What distinction, if any, is there between 'invented' and 
   'discovered' with regard to mathematics?

Although I agree that it makes no difference, for the sake of the 
essay I must come up with some logical arguments. I think that the 
concept of mathematics, as a whole, was discovered. I don't think that 
it could be invented. However, I can't put into words why I think 
this; it's very difficult to explain. What do you think?

3) Is it possible that 'invented' and 'discovered' are essentially the 
   same thing?

4) Goedel was mentioned in the discussion. What philosophy did most 
   mathematicians have when faced with this question?

5) Where can I find more information or background on mathematics and 
   philosophy?

Thanks again for your help,
Angela

Date: 01/03/2001 at 19:10:43
From: Doctor Ian
Subject: Re: Was mathematics invented or discovered?

Hi Angela.  You're very welcome.  
  
>Although I agree with your response that "the question 'Is math 
>invented or discovered?' is meaningless" I have no choice! I have to 
>write the essay anyway. 

Yeah, I understand that. I was just trying to give you some options 
for making the best of a bad situation. As a friend of mine likes to 
say, 'If life hands you a lemon, build a high-pressure explosive gas 
powered lemon gun and blast it to oblivion'.  :^D

In other words, when given an assignment like this, I've often had 
success analyzing the assignment itself, rather than just submitting 
to the tyranny of educational orthodoxy. But I realize that this 
approach isn't for everyone.

>1) What exactly is Peano mathematics?

Peano mathematics is a game defined by the rules described here:

   Peano's Axioms - Carl Lee, Dept. of Mathematics, Univ. of Kentucky
   http://www.ms.uky.edu/~lee/ma502/notes2/node7.html   

It's what you normally think of as 'arithmetic', but it's just one 
possible _kind_ of arithmetic.  

>2) What distinction, if any, is there between 'invented', and 
>   'discovered' with regard to mathematics?

I honestly don't know if there _is_ a distinction... or, more to the 
point, I don't know that you could ever prove that one exists. You 
can't really even begin looking for the distinction until you answer 
the prior question of what constitutes mathematics.

Is mathematics a process? Is it a collection of facts, or patterns, or 
theorems? Would mathematics continue to exist if all conscious beings 
were eliminated from the universe? Do the theorems that we haven't yet 
worked out already exist, in some implicit form? Or do they come into 
being only when they become known?

This is why I think you should read 'A conversation with Einstein's 
brain'. It's pretty short, wonderfully written, and it will give you a 
_lot_ to think about with respect to this issue. (That way, regardless 
of how the essay comes out, the experience won't have been a complete 
waste of your time.)

There is a great scene in Kurt Vonnegut's book, _Cat's Cradle_, in 
which a hard-core scientist is arguing with his secretary about truth. 
He challenges her to say something that is absolutely true, and she 
responds by saying "God is love." He looks at her and asks, "What is 
'God'? What is 'love'?"

This is the essence of philosophical exposition, as practiced in the 
West. You choose two words whose definitions aren't agreed upon by 
anyone, and then proceed to claim that there is some necessary 
relationship between their referents:

     Math is [invented | discovered].

What is 'math'? What is 'invented'? What is 'discovered'?

>Although I agree that it makes no difference, for the sake of the 
>essay I must come up with some logical arguments. I think that the 
>concept of mathematics, as a whole, was discovered. I don't think 
>that it could be invented. 

What is 'invented'? What is 'discovered'?  :^D

>However, I can't put into words why I think this; it's very difficult 
>to explain. What do you think?

I think you should read Richard Mitchell's book _The Gift of Fire_, 
see:

   Richard Mitchell's The Gift of Fire
   http://www.sourcetext.com/grammarian/gift-of-fire/   

I think you should read it _today_. (It's online, so you don't even 
need to go find a copy.)

>3) Is it possible that 'invented' and 'discovered' are essentially
>   the same thing? 

That's an excellent question. If there is an essential difference 
between 'discovery' and 'invention', would it perhaps be that 
something discovered has an independent existence, while something 
invented does not? Something uninvented doesn't exist, but something 
undiscovered does? (What is 'exist'?  What is 'is'?)

Are books invented or discovered? I could write a computer program 
that, given enough time, could generate every possible sequence of 
characters, so in a sense, every possible book that could be written 
is just waiting to be found, right? What is the _essential_ difference 
between a book that I haven't yet found because my program hasn't been 
running long enough, and a reservoir of oil that I haven't yet found 
because my drill hasn't been running long enough?

Could it be that discovery implies physical existence? That would seem 
to be a convenient way out, but math _books_ have physical existence, 
and the patterns of neural activity that are formed in your head as 
you think about mathematics have physical existence. So you're not 
going to get off _that_ easily.  :^D

If you want to finesse the question completely, you might argue that 
mathematics is the process of inventing formal systems, in which 
theorems are discovered. I say that this 'finesses' the question, 
because it utterly fails to address the prior question of whether 
those formal systems are somehow guaranteed to emerge, given the 
structure of brains, in which case the systems themselves are actually 
discovered, rather than invented - but your teacher might not notice 
that.

There is one other small item that you should definitely read. Find a 
copy of the book _Surely You're Joking, Mr. Feynman_, and read the 
chapter called... I think it's called 'A Map of the Cat'. Check the 
index of the book for 'Whitehead' or 'essential object', and you 
should find the part I'm thinking of. You'll see why I'm recommending 
it as soon as you've found it.

(When you're done with the essay, go ahead and read the whole book.)

>4) Goedel was mentioned in the discussion. What philosophy did most 
>   mathematicians have when faced with this question?

Most mathematicians have never considered this question, except 
possibly when drunk, or at freshman bull sessions in college. They're 
too busy inventing mathematics to worry about the possibility that 
they might actually be discovering it instead... or too busy 
discovering it to worry that they might really be inventing it. I 
forget which.

>5) Where can I find more information or background on mathematics and 
>   philosophy?

If I were you, I'd probably go to the Google search engine at:

    http://www.google.com/   

and search on the keywords 'math philosophy'.

Reuben Hersh has written a book called _What is Mathematics, Really?_ 
that you might find helpful, at least as a source that you can quote. 
(Quotations count towards the 1000 word minimum, right?) He takes the 
'Platonist' view, which is the idea that "mathematical entities exist 
outside space and time, outside thought and matter, in an abstract 
realm independent of any consciousness, individual or social."

But if you want to take that idea seriously, you have to be willing to 
accept a pretty odd definition of 'exists'. (By his definition, 
unicorns 'exist' too. In fact, by his definition, everything exists, 
even if it doesn't.)

My personal opinion is that Hersh doesn't sufficiently appreciate the 
difference between 'existence' and 'implication', but that's something 
you have to decide for yourself.

And, as I suggested, you should really read James's book _Pragmatism_.

Good luck with your essay. If you're finding what I'm saying at all 
helpful, please, feel free to continue our discussion. I'm rather 
enjoying it.  

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/   

Date: 01/04/2001 at 02:45:18
From: Angela
Subject: Re: Was mathematics invented or discovered?

Hello Dr. Ian:

Again, thank you so much for your quick response. I was delighted to 
receive another message so soon. By the way, I agree with what your 
friend said about the lemons. I am somewhat taking the advice, while 
still operating within the boundaries of the requirements for this 
particular essay. I've found that my teacher rather enjoys analysis of 
any kind, related to the topic or otherwise. But unlike you, I feel I 
must partially submit to this "educational orthodoxy"!

Your response generated a lot of ideas that I can use in my essay. I 
especially enjoyed your suggestion to "finesse the question." In fact, 
I think that I will incorporate that into my closing paragraph.

Upon doing more research, I discovered that I must stand on the 
so-called "absolutist" side of the argument. I think that mathematics 
is universal and certain. Mathematics is discovered by the intuition 
of the mathematician, and then established in a proof. Do you consider 
yourself a mathematician? What do you think about this viewpoint? I 
also think that mathematics is woven into practically every aspect of 
the world, right down to nature. I'm sure you are familiar with 
Fibonacci numbers... however, I'm confused as to the exact name of the 
sequences, related to the Fibonacci numbers, that occur in nature. 
Any ideas?

The opposing view to the absolutist is the "fallibilist" who argue 
that mathematics is invented and is revisable and changing. They say 
that mathematical truths are invented, or are by-products of 
inventions. Although I feel this is partly true, the concept of 
mathematics as it relates to everyday application, was not invented. 
What do you have to say about this?

You commented:

>Is mathematics a process? Is it a collection of facts, or patterns,
>or theorems? Would mathematics continue to exist if all conscious 
>beings were eliminated from the universe? Do the theorems that we 
>haven't yet worked out already exist, in some implicit form? Or do 
>they come into being only when they become known?

I think mathematics is all of these things, and that it would continue 
to exist even without conscious beings. The theorems exist as well, 
they just need to be discovered... they are realized when they become 
known, but in actuality they are waiting to be found. Is this wrong? 
What have you heard in your experience with mathematics in general?

I have not yet read 'A conversation with Einstein's brain'. But I will 
before I turn in my essay.

Furthermore, what do the words 'invention' and 'discovery' mean to 
you?

Were there any mathematicians who where also philosophers?

I am taking into careful consideration everything you say. All this 
information has provided me with a deluge of topics to discuss in my 
essay. Thank you for all the suggestions. I will try to locate the 
books you mentioned.

Thanks,
Angela

Date: 01/04/2001 at 15:33:53
From: Doctor Ian
Subject: Re: Was mathematics invented or discovered?

Hi Angela, 
  
>I agree with what your friend said about the lemons. I am somewhat 
>taking the advice, while still operating within the boundaries of the 
>requirements for this particular essay. I've found that my teacher 
>rather enjoys analysis of any kind, related to the topic or
>otherwise. But unlike you, I feel I must partially submit to this 
>"educational orthodoxy"!

I understand completely, and I sympathize. It's certainly much easier 
to take my view of things once the whole matter of trying to get into 
a top university is no longer looming in the future.  :^D

>Upon doing more research, I discovered that I must stand on the so-
>called "absolutist" side of the argument. I think that mathematics is 
>universal and certain.  

You might want to look into something called the Axiom of Choice, (an 
introduction and links collection by Eric Schechter of Vanderbilt 
University) at:

   http://www.math.vanderbilt.edu/~schectex/ccc/choice.html   

Although the reference I'm citing here indicates that the axiom is 
universally accepted, I can tell you from personal experience that 
there are many mathematicians who feel that the axiom of choice has no 
place in mathematics, and refuse to accept any theorems that can only 
be proved by using the axiom.

So what this means is that right here on earth, we have at least two 
sets of mathematicians who disagree on how mathematics may properly be 
carried out.

Wouldn't it be a failure of imagination to assume that all 
mathematicians elsewhere in the universe would necessarily fall into 
one of these two camps? 

Animals, for example, may take a different view of things. Read, for 
example _What the Tortoise Said to Achilles_ by Lewis Carroll at:

   http://www.ditext.com/carroll/tortoise.html   

>Mathematics is discovered by the intuition of the mathematician, and 
>then established in a proof.

Sometimes. One of the things Goedel showed is that in any internally 
consistent formal system that can be created, there will always be 
statements that are true, but that are not provable within the system.

So, does that mean that these unprovable truths are somehow outside of 
mathematics? Or does it mean that your view of what constitutes 
mathematics may be too limited?

But more to the point, why is it 'discovery' when the intuition of a 
mathematician comes up with a new way of doing things, but 'invention' 
when the intuition of an engineer does the same thing? Why wouldn't it 
be just as correct to say "Mathematics is invented by the intuition of 
the mathematician, and then established in a proof."

If you want to conclude that theorems are discovered rather than 
invented, then to be logically consistent you would also have to 
conclude that endgame analyses in chess are discovered rather than 
invented. But that seems like a silly conclusion, doesn't it?

By the way, from the way you've phrased your statement, it sounds like 
you're using the word 'mathematics' to mean the _theorems_ that are 
produced by mathematicians, rather than the process of deriving those 
theorems. Is that correct? If so, you need to explain how '1 + 1 = 2' 
and '1 + 1 = 1' can both exist within 'mathematics'.

The answer, of course, is that although the two statements use the 
same symbols, the symbols have different meanings in the two 
statements. But that would imply that the meanings attached to 
mathematical symbols are arbitrary. And if that's true, how can math 
be 'discovered'?

>Do you consider yourself a mathematician? 

I consider myself a scientist. 

>What do you think about this viewpoint? 

As I said earlier, I think mathematical absolutism does violence to 
the concept of existence by failing to distinguish between existence 
and implication. We have the word 'implied' to describe a class of 
concepts that might someday exist but do not yet exist. It seems silly 
not to use it.

>I also think that mathematics is woven into practically every aspect 
>of the world, right down to nature. I'm sure you are familiar with 
>Fibonacci numbers... however, I'm confused as to the exact name of 
>the sequences, related to the Fibonacci numbers, that occur in 
>nature. Any ideas?

Math is a framework for creating models. Sometimes the models have 
close correspondences with the objects and events that we observe in 
the world, and sometimes they don't. But it's crucial to realize also 
that the models we create determine which objects and events we 
observe!

You've surely heard the joke about the drunk who looked for his car 
keys under a lamppost a block away from where he dropped them, 
'because the light is better there'. In addition to its primary nature 
as a game, math provides a lamppost under which scientists search for 
theories to describe the world. But the only theories that they're 
going to find will be the ones under the lamppost.

To claim that mathematics is woven into every aspect of nature is a 
little like looking at a car seat and concluding that every part of a 
car is made from leather.  

>The opposing view to the absolutist is the "fallibilist" who argue 
>that mathematics is invented and is revisable and changing. They say 
>that mathematical truths are invented, or are by-products of 
>inventions. Although I feel this is partly true, the concept of 
>mathematics as it relates to everyday application, was not invented. 
>What do you have to say about this?

I think that's _one_ opposing view. Personally, I'm still trying to 
figure out what the essential difference between 'invented' and 
'discovered' is. I haven't been able to identify one.

>You commented:
>
>>Is mathematics a process? Is it a collection of facts, or patterns, 
>>or theorems? Would mathematics continue to exist if all conscious 
>>beings were eliminated from the universe? Do the theorems that we 
>>haven't yet worked out already exist, in some implicit form? Or do 
>>they come into being only when they become known?
>
>I think mathematics is all of these things

If you think that mathematics is a process, and if processes are, by 
definition, arbitrary, then how can mathematics be discovered?

>...and that it would continue to exist even without conscious beings. 
>The theorems exist as well, they just need to be discovered... they 
>are realized when they become known, but in actuality they are
>waiting to be found. Is this wrong? 

Whether it's 'wrong' depends on how you define 'exists'. If a theorem 
that hasn't yet been articulated can 'exist' in the absence of any 
being that can possibly understand the theorem, then I no longer know 
how to complete the following sentence in a sensible way:

     _______ does not exist.

Which is to say, 'exists' ceases to have any meaning at all. To say 'x 
exists' conveys no more meaning than to say that 'x zorbles' or 'x 
blinmarges'.  

But perhaps you can find something to put in the blank? (Be careful, 
because as soon as you stick something in there, I'm going to say that 
it 'exists' in some realm independent of space and time and physical 
causality.)

I'm not sure whether he ever really said it, but President Nixon was 
often quoted as saying that the United States might have to 'destroy 
Cambodia in order to save it'. I think that the absolutist position 
destroys the concept of existence in order to 'save' it - that is, in 
order to claim that it applies to mathematical concepts.

>Furthermore, what do the words 'invention' and 'discovery' mean to 
>you?

As far as I can tell, the only practical difference is that you can 
patent inventions, but not discoveries. But that's a legal 
distinction, not a semantic one. I'm not sure that there is a 'bright 
line' (to borrow a legal term) that can be drawn between the two.

If I wanted to try to create a hard distinction, I might try this one: 
Discovery applies to situations in which you don't get to choose the 
rules, while invention does not. So you can discover a law of nature, 
but you must invent a new technology in the context of other 
technologies that have already been invented.

Another way to say this would be to say discovery is context-free, 
while invention is not.

If you accept this definition, then the answer to the 'invention or 
discovery' question depends on whether cognition is context-free - 
that is, whether there is only one set of rules governing all possible 
modes of thought. Which hardly seems like a simpler question!

(It's like the old joke: How do you put four elephants in a 
Volkswagen? Two in the front seat, and two in the back!)

And even if the answer to that question turns out to be 'yes', then 
you also have to consider whether cognition is inevitable, or 
accidental.

So I guess I would say that if there is only one set of rules 
governing cognition, and if the rules of the physical universe make 
cognition inevitable, then math is discovered. Otherwise, it is 
invented.

But I might be able to talk myself out of that position.  :^D

>Where there any mathematicians who were also philosophers?

Richard Feynman, for sure. And Archimedes.  

>I am taking into careful consideration everything you say. All this 
>information has provided me with a deluge of topics to discuss in my 
>essay.

I'll bet '1,000 words' never seemed so small before!

As I said, I'm enjoying this. If you'd like to talk about any of the 
books, feel free to write back. Good luck with your essay. I'd love to 
read it when you're done.
 
- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/