Science is the study of observable reality, where we can observe physical objects. Maths has no physical observable reality, it is merely a method of understanding the world, created by the human mind. It is a faith system upon which we have come to rely upon for understanding our World.

In the year 2,000 I had a West Papuan student live with me for six months, his family and tribe was from a mountainous area who had learned to survive on basic things in their environment. They only had specific terms for up to three objects, so while he was with me he set about creating a mathematical system for them to learn maths in their language. The term one is not a physical reality it is a title given to a single item, but the item is not called one.

Maths like religion is merely a way of humans interpreting the Universe. It is a faith system.

Neither.

https://en.wikipedia.org/wiki/Mathematics

Dear Josephus,

Well my friend there was a hell of a lot of Math involved in getting a man to the moon. I might need you to tell me what parts they were content to leave up to faith.

Dear Josephus,

If you had been to university (and studied something other than one of the trade degrees) you would know the answer to that question.

Check this out

https://curiosity.com/paths/is-math-a-feature-of-the-universe-or-a-feature-of-human-creation-idea-channel-pbs-pbs-idea-cha

Jospheus there is no definitive answer to your question.

Some mathematicians/physicists due actually think that the universe is maths (not just described by maths but actually *is* maths-- some ancient Greeks had this idea as well). Search for the "Mathematical universe hypothesis" by Tegmark and also "Mathematical Monism" for more.

While others think that it is just a purely a computational construct and that it is just boring/meaningless symbolic manipulation according to set rules. People who do "maths" just agree on a set of rules to play by and then make "moves" according to those rules. For them it is just pure fluke that it can be useful at all in describing the real world (whatever the "real world" is). If I remember rightly Hilbert (a very famous mathematician of the 19/20th centuries) was of this opinion and said something that is commonly quoted that explains this position (can't remember the quote but it mentions tables and beer mugs, google would find it if your interested).

Yet others say that maths is an empirical study and that we can only know maths by observing the universe by empirical study like any other science.

There are even other schools of thought about maths. Wikipedia has an article on the philosophy of maths that lists various approaches/schools of thought. It's been a while since I've seen it but it used to list at least 5 different philosophical view points.

ps-- I myself am tending towards intuitionism/constructivism.

I must ask my sister-in-law what her view is as she has a PhD in Maths and a lecturer at University teaching teachers of Maths. She has devised several ways of arriving at the same conclusion by using different systems. Maths is a theory of understanding reality.

"In the year 2,000 I had a West Papuan student live with me for six months,"

Well was it really six(whatever that means) months or do you just have faith that it was more that five(whatever that means) months.

And let's not even get started on the total unreality of 'months', 'years' and 2000!

mhaze,
Actually it was 5 full moons, as their ancestors knew nothing about months or their Roman names, which is our system of measurement of time. Who said there were 24 hours in a day, why aren't there 100 increments in a period from the furthermost point of the sun shining on us to the next. Yes time is a handy human invention in which we are indoctrinated.

//Who said there were 24 hours in a day, why aren't there 100 increments in a period from the furthermost point of the sun shining on us to the next. Yes time is a handy human invention in which we are indoctrinated.//

I believe it was the Babylonians who gave us the 24 hour day. They used a base-12 counting system, which is why there are 360 degrees to a circle, 60 minutes to an hour, 24 hours to a day and so forth.

There was a push, at one point, to adopt a metric clock. The metric system was such an improvement over previous systems when it came to currency, weight and length measurement that it seemed obvious to some people that time should be measured similarly, but it never really caught on.

These days a second is defined as the time that elapses during 9,192,631,770 cycles of the radiation produced by the transition between two levels of the Caesium-133 atom. The time it takes for a cycle of the radiation produced by the transition between two levels of the Caesium-133 atom is fixed, which is why it forms the basis for the definition of a second, but the choice of 9,192,631,770 cycles is purely arbitrary - we could make the second longer or shorter by increasing or decreasing the number of cycles used. This would not have the effect of making time pass any faster or slower in your frame of reference. Humans did not invent time, just ways of measuring it.

Maths is a human invention, just like any religion or God is.
The humans that wrote the bible believed the earth was flat, but that certainly didn't make it so.
Mind you, I would put my trust in the science of mathematics any day, rather than any belief in invisible gods...

Suseonline,
Well it is a beginning that you believe in the invisible human invention of maths, while still denying their is design in the Universe that can be rationally accepted as deliberately placed there by a designer.

Of course mathematics is not a belief system. Only someone without a knowledge of history, sociology and anthropology would believe that it is a belief system.

My sister in law calls Maths a language as she has a PhD in Mathematics and teaches several languages of Maths to arrive at the same conclusions.

//The humans that wrote the bible believed the earth was flat, but that certainly didn't make it so.//

They probably didn't. The ancient Greeks knew full well that the Earth was spherical; Eratosthenes of Cyrene measured it's circumference to an impressive degree of accuracy in 240BC. The New Testament was written in ancient Greek and it is generally accepted that the early Christian church held largely to a spherical geology. St Augustine of Hippo, for example, knew that the world was a sphere.

It think a lot of people get a bit confused about this one because the Church clung to the geocentric model of the solar system for centuries, even in the face of overwhelming evidence and famously persecuted Galileo on the pretext of his support for the heliocentric model.

//Mind you, I would put my trust in the science of mathematics any day, rather than any belief in invisible gods...//

It's not really a science... mathematicians don't base their theorems and proofs on observations and there are no 'experimental mathematicans'. I'd say that in some ways it is closer to the arts than the sciences, even if scientists do use a lot of maths.

I do agree that I'd rather have civil engineers designing bridges using maths than relying on the power prayer hold them aloft. Maths is useful and practical (although it can be fun and recreational as well), whereas religion doesn't really seem to have any useful purpose other than making the runners of this world feel smug and self-righteous. Oh, and inspiring murderous acts of religious fanaticism... yep, that seems to be about it.

"Maths like religion is merely a way of humans interpreting the Universe. It is a faith system."

No it isn't, unless you are misusing the denotations and connotations of the word 'faith' as they apply to mathematics and to religion.

Claims in mathematics are subject to proofs which establish the truth of statements.

Claims in religion are are subject to statements to establish their truth instead of proofs.

Mathematics is a creation of the human mind. As one studies the relationship of mathematical entities one is filled with beauty and awe. An elegant proof can evoke the same sort of reaction in the mind of a mathematician that a Bach score will evoke in a trained musician.

We define a point as a location in space. It has location but no non-zero dimension. A point is something we can never see (an abstraction which cannot exist as a visible object), but we place a dot to represent it. There are such marvellous mathematical concepts as dualism which indicate that point and line can be interchangeable and still make sense in geometrical axioms and theorems. Two points determine a line, and two lines determine a point.

Mathematics is sometimes taught so one can think it is no more than numeration, but set theory and topology for example do not have to have anything to do with numbers.

Whether math is a science or a human belief system depends on the definition we give to a science and a human belief system. We can define either a science or a human belief system in such a way that mathematics fits the definition. However, mathematics is to me a triumph of the human spirit - a soaring, wondrous art.

Not everyone can make mathematics - create new branches, state new axioms and postulate new theorems. However, almost everyone can be educated to appreciate mathematics. It has the beauty that most of us can understand a mathematical conjecture, but a mind of great subtlety and depth is required to prove the conjecture is true.

An example is the Goldbach conjecture. Every even number (except 2) is equal to the sum of two prime numbers. No one has found an exception, and no one has proved it is true for all even numbers.

Dear david f,

What a load of gobbledygook. Mathematics is mathematics no matter how complicated you try to make it. It's simply just all about calculating and measuring the world (and universe) we live in. My first degree was in engineering and it was just an exercise in applied math and I was glad to see the last of it when I graduated.

I have been silent so far because the question of mathematics is difficult and I don't have a definitive answer.

However, I cannot remain silent on the repeated claims as if religion has anything to do with understanding, rightly or wrongly, the physical world that we live in.

Unlike psychoanalysis which endlessly attempts to understand and assign meaning to our dreams, the object of religion is to awaken from this nightmare that we call the world so that we can "remember" who we really are - God.

The confusion arises because historically, some people who claimed to be religious and to teach religion, were ALSO interested in understanding the world around us; and unfortunately they were believed because they taught it with the same authority. It is quite possible - and I've seen it several times myself, for someone to be a great authority on religion, yet prattle nonsense when it comes to describing the world (and the opposite is also all too common).

Dear Mr Opinion,

I'm sorry your math teachers were inadequate. Mathematics is much more than calculating and measuring. A blind person may not be able to appreciate a sunset, but a blind person can appreciate mathematical relationships.

Some blind mathematicians:

Leonhard Euler - Swiss mathematician and physicist who went almost totally blind at fifty-nine, but his productivity on mathematics did not decrease throughout his life.
Bernard Morin - topologist from France.
Lev Pontryagin - Soviet mathematician who went blind at fourteen. He continued mathematical study with the help of his mother Tatyana Andreevna, and made major discoveries in a number of fields of mathematics.
Nicholas Saunderson - English mathematician who went blind at the age of twelve months, held in high esteem by Isaac Newton.

They were not blind to the beauty of mathematics

Well what can I possibly add to those, who've eloquently shared their opinion on this fascinating subject ? Unfortunately I could never quite understand many of even the most basic mathematical concepts I tried so hard to grapple with at school, I simply couldn't 'get it' as the say ?

My first two years in High School, I was required to undertake Maths l & Maths ll, and ultimately my teachers must've finally realised I was essentially an utter dunce, mathematically at least. So in third year I was fortunate enough to be permitted to undertake General Maths studies instead. Consequently I managed to successfully pass in that subject, in the 1956 Intermediate Certificate Examination which at the time, was an important scholastic objective, back in the fifties and sixties.

Today, I honestly believe you can either 'do Maths' or you can't ? In which case, you just have to accept that you're not all that bright. Perhaps the only vocational avenues that may be available to you, post your schooling, are a railway fettler, general labouring, become an Infanteer in the Army, or as an absolutely last straw, join the police force ? And you know what they say about police officers, '...too lazy to work; too frightened to steal; and haven't got the brains to get a satisfactory job...' ? I didn't do too badly I suppose ? I participated in two of the four options mentioned above - Australian Regular Army and the Police Force.

Dear david f,

I didn't say anything about having inadequate teachers. It appears that your knowledge of English is as weak as your knowledge of mathematics. Just out of curiosity, what fields do you hold your degrees in?

Dear Mr Opinion,

You didn't say anything about having inadequate teachers. I assumed that your lack of appreciation for the beauty of mathematics was due to that. Of course it could be due to something else.

My degrees are in mathematics and physics.

Being an electrical engineer, and having done both pure maths, applied maths, physics and engineering, I found that mathematics was an important tool to describe or understand pretty much everything scientific.

As one of my professors once said, applied maths is the practical application of mathematics, Physics is applied maths applied to the physical world, and engineering is applied physics.

Dear david f,

My four degrees are in engineering, anthropology, sociology, and history.

Also engaging with Shadow Minister's subsequent comments I would like to offer the following definitions:

Mathematics is the system used to calculate and measure our world or universe.

Physics is the science used to understand how our world or universe works.

Engineering is just a vocational degree (like medicine, architecture, law, etc.) with the singular purpose of training people for specific jobs.

Dear Mr Opinion,

You defined mathematics as 'the system used to calculate and measure our world or universe.'

Your definition of mathematics is limited since it does not cover all the branches of mathematics. Calculating and measuring are only two of the functions of mathematics. Topology and number theory neither calculate nor measure but are branches of mathematics.

A definition from the net is:

the abstract science of number, quantity, and space. Mathematics may be studied in its own right (pure mathematics), or as it is applied to other disciplines such as physics and engineering (applied mathematics).

Dear david f,

I'll stick with my definition of mathematics. It may be simplistic but it is a good basis for understanding the origin of mathematics amongst the ancients and it's subsequent evolution toward modernity.

I definitely do not agree that physics is applied mathematics and thus some sub-branch of mathematics. Physics stands by itself as the branch of science that studies and analyses the way the world or universe works.

Engineering definitely is not an applied mathematics. For me engineering is only a trade or profession that in many respects can exist independently of mathematics. I think engineering faculties complicate the study of engineering with an overuse of mathematics in order to elevate what they do to the intellectual level enjoyed by the Sciences and Arts.

Dear Mr Opinion,

You wrote "I'll stick with my definition of mathematics. It may be simplistic but it is a good basis for understanding the origin of mathematics amongst the ancients and it's subsequent evolution toward modernity."

A definition of mathematics should define mathematics. Your definition doesn't. However, 'it's' is an abbreviation of 'it is'. The possessive case of it is its. English, unlike mathematics, is not logical

Dear david f,

Oops! Thanks for pointing out my typing mistake. I should check my typing before uploading my comments.

Dear Mr Opinion,

I appreciate your polite response to my criticism. It makes it pleasant to argue and/or discuss matters with you.

The answer is "who cares?!"

Many things in this world are irrelevant.

Like for example I don't need to know the science behind how a microwave works in order to cook food, or I don't need to know how the internal combustion engine works to drive a car.

Or other silly questions like the question of multiple universes, questions that no-one will ever know the answers to.

Just a pointless use of brain power.

The most important thing you need to learn about maths is that 2+2 is ALWAYS going to equal 4.
It's never going to change, and don't let anyone ever try to teach you different.

Armchair Critic: Mathematicians, Logicians and Philosophers care. But not only them, I dare say that almost everyone would care in certain circumstances. For example, if someone proved tomorrow that 1+0=0 then all employers could refuse to pay their employees because by using this sum it is trivial to prove that any number equals zero, ie: if they owe $1000 dollars to an employee then by manipulating this sum they can show that they really owe $0.

Now this may sound stupid to you, to think that mathematicians could prove a sum like this, but this has actually happened. A bit more than a hundred years ago Bertrand Russell discovered a flaw in mathematics. It was a very, very low level flaw, at the foundational level of set theory. (For most mathematicians, set theory is the lowest level of maths and all results of maths are constructed from set theory. Numbers and operators like addition are actually sets, but very,very complicated ones). Now, any flaw in math, no matter how small, allows you to prove results like the above. See: Russell's paradox on wikipedia for more info. This flaw shocked the mathematical world at the time and people straight away searched for ways to get rid of it. They achieved this by changing the axioms of set theory, that is, they changed at a foundational level what we consider to be true and not true. By doing this, they laid bare for all the world to see that the base axioms and rules of math are arbitrary, you can change them as you see fit. That is, maths can viewed as just a belief system in which you choose what you believe.

Hey thinkabit,

I did pretty well in maths at high school and was placed in an advanced maths class, but I played up in class and they put me back into the normal class because I wasn't applying myself.
I didn't engage in any formal studies after high school Grade 10, so trying to understand Russel's Paradox and this 'set theory' stuff is a little above my pay grade.
- I don't understand what the strange looking symbols mean.

For me though, things don't need to be so complicated like I said earlier.
Take your example- 1 + 0
Lets say 1 = Something;
and 0 = Nothing

You cant start with something, add nothing, and have a total of nothing, its impossible.

2+2 will ALWAYS equal 4.
To entertain the idea that adding 2 and 2 can result in any number other than 4 is to me absurd, like 'you are now entering the twilight zone' absurd.
(So you we're right - it does sound stupid to me)

But I must say I wasn't aware of Russel's paradox before you mentioned it and I'm glad you did, as I found this trivial little bit of learning curiously interesting.
Now I understand where people get this 2+2=5 stuff from.

For me though there are no flaws in maths.
I like the term 'mathematical certainty'.
So I'm thinking there must be a flaw in this 'set theory' instead.
I don't think maths should be considered a belief system in which you choose what you believe.
I don't even know why they would try to teach people that, but hey, I'm not a Mathematician, Logician or Philosopher.

Thanks for the info.

Kurt Gödel's incompleteness theorems established that there is uncertainty in mathematics.

https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems

Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic. The theorems, proven by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The two results are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible, giving a negative answer to Hilbert's second problem.

The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an "effective procedure" (i.e., any sort of algorithm) is capable of proving all truths about the relations of the natural numbers (arithmetic). For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system. The second incompleteness theorem, an extension of the first, shows that such a system cannot demonstrate its own consistency.

Hello davidf,
I was going to mention Godel's theorems in my post above, but decided that it would complicate the post. It does fill out/complete the story of people searching for a way to remove Russell's paradox from maths.

Hello ArmchairCritic,
You seem absolutely certain that you "know" there are no flaws in maths. The question I have for you is: How do you know?
If you say something like "Because I just know that it is true" then you've basically admittedly that it is something you believe purely by blind faith. Believing something is this way is no different from a Muslim saying that he knows that the Koran is the inspired word of god that we must submit to its teachings or a Christian saying that she believes that "Jesus is the way, the truth and the light and no man can enter heaven except by him". Ie, for you maths is a belief system taken on blind faith.

If on the other hand, you may say something like "Maths is true because whenever you look at the world around you, when you add 1+1 you always get 2, eg: one apple + one apple always gives you two apples.". In this case, you are saying that maths is a way of summarizing the results of empirical observation of the universe. That is, for you maths is a science and we find out about math by systematic experiment on the "real world".

Choosing one of these two views points as the proper view point is what the original post of this thread is about.

However, these two views don't encompass all current schools of thought on maths, there are other ways which you can answer this question. Read the Wikipeadia article on the philosophy of maths for more.

Dear Armchair Critic,

Not everyone is happy to go through life being an ignorant person. There are a lot of us who want to know about the world and universe we live in; and people like me who also want to know everything about humankind.

That is why we study the Sciences and Arts so that we don't have to go through life being ignorant. For people like you ignorance is bliss but for a lot of others ignorance is anathema to their very being.

Armchair Critic wrote:

"You cant start with something, add nothing, and have a total of nothing, its impossible."

In computers there are AND gates and OR gates. In an AND gate all inputs must have a signal or there is no output. In an OR gate if any input has a signal there is an output.

1 and 0 can represent a signal and no signal, respectively. If we have two signals coming in the gates the arithmetic is as follows:

OR gate

0+0=0
1+0=1
0+1=1
1+1=1

AND gate

0+0=0
1+0=0
0+1=0
1+1=1

davidf: To be fair to ArmchairCritic, you not doing arithmetic addition there, just propositional logic which happens to share the same symbols of "+", "1" and "0".

To illustrate your point better you could have asked something like this:

What does: 1 + -1 + 1 + -1 + ...(repeat forever) equal?

Does it equal 1? Since we can rewrite it: 1 + (-1 + 1) + (-1 + 1) + .... = 1 + 0 + 0 .... = 1

Or does it equal 0? Since we can rewrite it: (1+ -1) + (1+ -1) + ... = 0 + 0 + 0 + .... = 0.

Of course, a person who knows a thing or two (especially those who know about Cauchy limits and convergence) would say that I'm cheating here and are abusing the proper use of maths.

Dear david f and thinkabit,

It's good to see you using mathematics for its intended purposes: calculating and measuring.

Mathematics is a language with a great deal of internal logic- its neither a science nor a human belief system. Its totally context dependent- people who says it not just compare 10 grain of sand to 1 boulder.

I'm not ignorant.
I just know that 1+1=2.

Lets say one plus zero could equal zero.
How can you apply this miracle of backwards thinking in the real world?
What use is your madness to society?

Say I go to the bakery and buy a pie.
Then before I leave I decide maybe one might not be enough so I buy another one.
How many pies do I have?
Don't try and tell me 1, none or 3
You know what the answer is.

1+1 will ALWAYS equal 2.

Armchair Critic: As it stands, right now (today) there are no known flaws in standard arithmetic, well as far as I know anyway. By Standard arithmetic I mean arithmetic according to Peano's axioms built with ZFC set theory to make sets that form the things being talked about (whether these things be numbers, functions, operators such as addition, etc.), with first order calculus to express statements/theories about these things and some deductive system (such as a Hilbert system) to create proofs of these theories. This gobbledegook is a formal description of what you would most likely intuitively mean by arithmetic/maths.

However, just because there are no known flaws today, doesn't mean that someone won't find one tomorrow. In fact Godel's theorems guarantees that we can *never* be certain (certain within the framework of standard maths) that no flaws will ever be found.

So, to make a statement that 1+1=2 will *always* be true is to make a statement beyond/outside of maths. Ie: you are claiming something beyond which maths can prove. In other words you are relying on greater/separate power/source beyond maths to make this claim. An example of such a power is blind faith.

Now, if it is blind faith by which you make this claim, then your belief in the correctness of maths is no stronger than any other claims made on blind faith, such as those made by religious people. Note also, that I could make the counter claim that "maths is flawed" on blind faith (without a proof) and this claim would be just as valid as yours.

cloa513: Your example of comparing 10 grain of sand to 1 boulder doesn't add any weight to your claim that maths is neither a science nor a human belief system.

However, I don't dispute that maths certainly has a precise and (almost) unambiguous language intrinsically associated with it that is used to convey meaning between working mathematicians.

By-the-way maths has a branch specifically related to handling statements like the example you gave-- it's called type theory. In computer science type systems form a major part (in my opinion the most important part) of programming language theory. An everyday example of type theory in action is the common statement that "you can't add/compare apples to oranges". see: https://en.wikipedia.org/wiki/Type_theory for more info.

Additionally, it might interest you to know that maths includes the study of formal grammar systems as a field of study. Most people are surprised to find that this is actually a field of mathematical study since they normally associate grammar with English studies/Literature at school. In fact some grammar systems are extremely powerful mathematically- an important result of grammar systems theory is that there exists a grammar which is Turing complete: ie, there are grammars which can be used to "compute" *any* mathematical statement. see: https://en.wikipedia.org/wiki/Unrestricted_grammar