On The Nature Of Numbers

I write this in response to ongoing discussions about proving calculation techniques.

What is a number?

Most people have no clue how to answer. They might pick up a few coins and try to demonstrate how they use them, but they can’t explain what they are. It’s a fascinating fact that this question was not really tackled until the 19th century by men like Cantor, Cauchy, Dedekind… The mathematical development of number systems that include fractions and irrational is very technical and deep but the Natural numbers and then Integers are simpler so I will talk about those examples.

The modern Integers are a set of abstract objects whose nature depends entirely on their formal definitions. The Integers are given by the Construction of the Integers which are the rules that make them exist and govern their behavior.

Consider a close parallel, a game like chess, checkers, or card games. Then what is a chess piece? What actually is a Knight. The little figurine is superfluous. It’s just a notation convenience. It could be a piece of paper with a K on it. It need not have any physical form at all. When one plays online, there are no real pieces. Strong chess players can play without sight of the board. You don’t need a chess set to play chess - all you need is the rules and a brain to explore them. A chess piece’s nature, it’s very existence comes out of the rules of the game. A knight is entirely defined by its relationship to other chess pieces and the board. A knight is nothing more than a web of relationships with other chess pieces. *It is self contained. It makes no reference to the physical world".

Likewise with numbers, an integer’s properties, its existence is defined by its relationships to the other numbers. A web of relationships with nothing at the vertex. But numbers have value. There is something there surely - 7 has a value. But 7’s claim to a value is founded on it’s being a multiple of 1’s and one’s have a value of 1 because they are defined that way. The value is created by definition!. Ex Nihilo!

Might seem outrageous at first but money works the same way and we are quite comfortable with it. $1 has value by definition. It is not defined in terms of any other currency or anything else of value.

So what about their application to the physcial world? Chess has no direct correlation. Mathematicians construct endless numbers of these systems of abstract objects and play with them because they are fun, like chess. That’s pure mathematics. But we use numbers to think about the physical world. We use numbers as a model of things in the physical world. They allow us to make useful predictions. Instead of putting all the sheep in one pen to count them, one can figure, there were 21 sheep in that pen and 15 in the other - that’ll be 36 altogether.

But if one pen holds chickens and the other foxes, then numbers are not a good model for predicting what happens when you put them together. Sometimes things are close enough to be useful, with care. Quantities of money are not true numbers. The 1 in $1 is only divisible by 100! If I divide $1 by 300 I get pieces that are $0.00333. There is no such unit of money. $0.00333 is nothing. This is a real problem in financial transactions and a number of scams have been pulled off using this trick. If I am a bank holding your money, I can ‘process’ it by splitting it up into tiny fractions of a cent, round them all to zero and declare you have a zero balance.

When you learn to play chess you have to put aside what you know about horses and bishops. You think only about the rules and what can be done with them.

With numbers too, it can be treacherous if you use practical real world examples as your main understanding. The reason 1+1 = 2 should not be validated by a reference to some practical example, it should be explained by the rules of the game.

Edited- stupid error in one line.

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And here I take the opposition.

An Integer is truly a natural number. It counts a thing. It takes away a thing. It is the absence of a thing. Not merely an abstraction. An Integer is discrete and descriptive. It is grounded in reality and is a critical link between the pure abstraction of mathematics and the understanding of physics.

… and then again I could be completely wrong.

There is a pretty good introduction to the properties of numbers in Michael Spivak’s Calculus textbook that I am going to read again a couple of times this weekend. There is also a different but approachable description of numbers in Professor Klietman’s Calculus for Beginner’s and Artists. http://www-math.mit.edu/~djk/calculus_beginners. It seems like a central topic in many introductory texts.

Proving algebraic identities walks you straight into the philosophy hall if you let it. Don’t get me wrong, I love philosophers but I am too old to be unsure about my existence. The modern physicists that accept that reality is an abstraction because the math works need to give their head a shake. When 1=0 or you come up with an infinity you can be pretty darn sure that you have made an error. This is the nature of Integers to my mind.

I imagine there is some kind of hell for number theorists that is nothing but the number 1. The set of all sets contains itself. Hard to believe Russell only came up with this in 1901.

<\rant>

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I’ve been watching lectures on Ramanujan’s estimation algorithms from his notebooks tonight. There are quite a few that I can understand the explanation of the formulas for but for the life of me I cannot understand how he came up with them. I’m not sure if he simply picked some really entertaining constant or relationship and then said. Let’s make that equal pi or if instead he actually had a method for identifying unlikely relationships. oh look, a light, at the end of the tunnel… train.

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An integer exists without counting things. The integers make no reference to the outside world. There is nothing in the Construction that requires any physical existence at all. These are the numbers you are using when you do algebra. These are the foundation of the laws of the algebra that you do. You may believe that numbers are inherent in the nature of the universe but those numbers are not our numbers. We do not know what rules them and only guess that they behave like ours.

Their usefulness in counting reflects their usefulness as a model. It is not a requirement of the integers. The number you attach to a set of physical things is totally arbitrary and depends on the counter’s definition of his set. A flock of sheep has no inherent number. Were we counting just yearlings? Do we count the lambs as half? The ones ready to go market, their ears, the total number of wool fibers. This is completely chosen by the person who does the counting.

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We are getting philosophical here.

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“The Goddess puts the numbers on my lips” was all the explanation he would give for his insights. Much of his work was unproven or supported by inadequate proofs and it was his colleagues in England who labored to provide solid proofs for his ideas.

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Yes, of course. Philosophy is the study of ideas and the foundations of mathematics and it’s various objects like numbers, matrices, elliptic functions, differential equations are ideas. examining them puts you in the context of philosophy.

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Actually a ‘number’ as it is commonly referred to is simply a random symbol.
And, simply put, numbers represented, as a symbol, real world quantities.
These are now called natural numbers since the development of mathematics and not simply the need to keep track of how many sheep I have.

From my point of view, mathematics as we know it is a system of symbol manipulation through the rules of logic predicated on the real world since the symbols foundational representations were rooted in the real world.

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For clarification let me explain, I’m not trying to impose a ‘trip’. If you have strong personal feelings about the nature of numbers, I don’t challenge that. Believe as you will. I am a utilitarian. I make no claim to have the Truth. If my tone seems to be lecturing, I apologize. This is the way I talk to myself in my head and I don’t have another.

My point is that his is the most useful way to understand mathematical statements about algebra. The numbers and algebra we use, now follow the rules laid down by modern mathematical theory. They are sanitized of any references to the physical world. If you cling to real world analogies you may be led astray and you may have a hard time taking these ideas on board.

I have taught mathematics - at college level and I remember my own early struggles too. When people turn to physical examples to make sense of some of these ideas it leads to misunderstanding rather than insight. Yes it works with simple addition and apples but that’s about it. When you need a negative apple, you are done with that example.

If you try to understand chess using what you know about bishops and horses you will be badly misled. That’s an extreme example - the confusion is more subtle with numbers because they are designed to mimic the way we experience the physical world. They are designed to be useful. Chess doesn’t care about that. But because of that, misconceptions about numbers lie deeper and are more treacherous.

Legend has it that the Pythagoreans, a mystical cult who believed that the world was constructed from integers (they allowed fractions), threw one of their students overboard when he produced a simple proof that the square root of two cannot be rational.

The Greeks steered most of their efforts away from algebra, except for Diophantine Equations which deal with integer solutions. They were unable to accept or find a coherent way to deal with infinite sets because they couldn’t make sense of them in the physical world. Zeno’s Paradox is an example of this.

Numbers seem like a simple straightforward matter but on close examination, they become strange and difficult. The Real Number line is a very strange object. No matter how much you expand it, it has the same density. Completely packed with points. And yet no two points are adjacent. The Real numbers are based on the topology of open point sets which too are strange. Consider, what is the smallest Real number s > 0 ? There you have a sharp boundary with no edge points! I defy any one one to find such properties in the physical world. For most people, naive intuition is no help here, one must look to the rules.

For myself, as a student, this insight was a game changer. I had to find this out for myself but as a teacher I have been able to steer students around this rock and save them considerable frustration.

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I didn’t think my post portrayed a strong personal belief about numbers. Your original statement was along the lines of what is a number and that most people have no clue how to answer.

Whereas, I don’t think that’s true. Most people, I believe would say a number is a symbolic representation of a real world quantity, and they would be exactly correct, because that is what a number is.

I also disagree that integers are a set of abstract objects that ‘exist’. Rather, they are a set of symbols which can be manipulated by logical rules and those rules are grounded in the real world. How integers behave may be defined formally, but that formal definition is also based in the real world.

Indeed, all of mathematics is grounded in the real world. If mathematical results do not match with real world observations then there is something wrong with the mathematics, not the real world.

Mathematical spaces do not exist. They are formalized logical structures and systems of symbol manipulation.

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That comment was not a direct reply to yours. I thought your comment was a good one and not so easily answered. But I will get back to you.

I have never gotten the kind of answer you suggest unless it was someone with exposure to pure mathematics or philosophy. This seemed to hold true even when asking engineers and scientists. As I say, I’ve done a fair amount of teaching.Nor is this definition to be found for most of our history of the development of mathematics.

And this brings up something I intended to write in this thread:

The idea of numbers came very slowly to us. Counting was done long before numbers were available. Counting is a one to one correspondence between two sets. Take a jar of cookies and hand out one to each child. You gave out as many cookies as there are children. No integers needed.

In ancient Babylonia, the first abstraction was to collect tokens. They kept piles of tokens that represented the seal on the urns of grain that had been collected for taxes. This allowed the scribes to manage supplies without having to be in the granary. The individual seals were replaced by strokes on a clay tablet. Then uniform grouping occurred where 5’s or 10’s were crossed off and and these in turn became symbols for larger numbers. These refinements were hundreds of years apart.

The abstraction was slow. There seems to have no bridge of sudden insight into seeing the Natural numbers that any young child now takes for granted.

The Greeks as, I mentioned couldn’t develop the abstractions necessary to think about infinite sets. In fact it took a long time. Renaissance mathematician used infinite sequences in an ad hoc way without rigorous proofs.

Even Newton was uncomfortable with negative numbers and rewrote his equations to avoid them. Complex numbers were resisted too for intuitive reasons that were not valid.

On the other hand, there cultures where there are no numbers larger than three! I know of no examples of numbers being used in pre history.

It’s very hard to argue that our sense of numbers, which seems so natural, is really inherent and that our untrained intuition is going to guide us well. Look at how long it took to come up with the concept of 0.

In every age mathematicians themselves have had to struggle to free themselves of parochialism. It seemed so natural and obvious that parallel lines continue forever at the same separation. How could space be otherwise. Yet, when we finally allowed ourselves to examine this question, it turned out that there are indeed other possibilities and one of them is a better explanation of our universe than the one Euclid had.

If the mathematicians themselves struggled to discover and accept these concepts, many of which we now use freely, perhaps it’s worth checking the baggage we have on board when we, as students set out on the journey to really understand them.

Yes Zvuv. Sorry. I was texting, so tended to be terse.

I suppose my response was a response to your post, but in the sense of some of the ideas presented, not particularly you yourself. I do agree that mathematics took a long time to develop, by I’m assuming that one was asking what a number was to an individual living in this time.

As previously mentioned, I don’t have a connection to numbers unless you mean that venerable branch of mathematics: numerology :wink:

Thank you for the kind comment regarding my post. I’m not really trying to argue much since one can’t really have these types of in-depth discussions on a thread, one can only dip here and there which I do think is sufficient though. I suppose I’m just bringing in some points to balance the discussion from the ‘man on the street’ perspective.

So to that end, I would also suggest that negative integers are part of the mathematical structures/rules which make things convenient. Negative integers are only negative with respect to some predefined reference such as occurs in the measurement of temperature. So rather than negative apples, one can have negative temperatures and have it mean something real, but only in terms of a predefined zero.

Of course there are no negative temperatures in nature as things get colder, only positive ones. Thus -10C, is actually +14F, and about 263 degrees Kelvin (the scale based on nature rather than on convenience). One cannot have negative Kelvin temperatures since this would mean a temperature colder than absolute zero where all molecular motion (the physics definition of heat) stops. So in this case the negative integers describing how cold it is have no real meaning either in reality or in mathemetics. (Lets ignore those who use negative Kelvin values to describe the extremely hot), since that isn’t with regard to ‘coldness’.

I agree that intuition will not help us plumb the depths of mathematics much since it is an extremely large and and extremely complex field. But I do think we have an intuitive sense of number as it pertains to living with nature.

Hi everyone,

And I take the 3rd non-discussed stance…Gödel’s theorem:

According to his theorem, any mathematical “system” is self refuting.

Simplified Gödel’s theorem? yes but I prove the point.

Stefos

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That’s a misstatement of Godel’s Theorem and irrelevant to the point of the thread.

I welcome discussion and dissent and correction if it is thoughtful. I get a bit impatient with those that make off the cuff rejoinders on a subject that needs some thought. I took a little bit of time to think about what you wrote. Also, I use it as a foil to write some things I intended to say, which is not completely fair. So I apologize for the length and the tardiness.

Your description certainly applies to mathematics but it’s very generic. That definition applies to logic and even to philosophy where philosophers often use symbols to stand for propositions. But it misses the point. It’s like describing writing as the business of typing while following a very complicated set of rules. Mathematicians definitely do shuffle symbols around but it’s not aimless.

Numbers have their roots in utility. In the beginning we counted concrete things. From there we developed some tricks of manipulation. The modern Integers have been carefully defined so that they perform this function well and arguably that is their primary purpose. But numbers do not have to represent real world things or any things at all other than multiples of 1 which itself is only a ‘thing’ because it is defined so. They now have an existence that is divorced from any application. If you read the Construction of the Integers, nothing is required to exist other than a brain to interpret the rules. You can have Integers without the Universe. That doesn’t mean that we can’t use them to represent real world quantities and that still remains our main interested in them but that is not a basic requirement. That’s the motivation. Numbers are not determined by their applications. Teleological Fallacy.

It’s a mistake to assume that because one thing grew out of another that it is still bound to its origins. The Genetic Fallacy. Mathematicians have long explored numbers for relationships that had no apparent utility. Fermat’s Last Theorem was one such. For a long time it was thought that Number Theory had no practical application until encryption technologies showed up. And they’ve gone on to create all sorts of number like abstract objects and play with them.

Counting, measuring length, weighing or measuring duration are operations that do not require numbers at all. In old wood shops, in the days when rulers were expensive, the craftsmen marked their dimensions on a ‘story stick’ and then hold it up against the new piece to mark it to the same length. I still work this way. Weighing likewise can be done just by balancing against a standard set of weights. Time can be measured by the sun, by a burning candle. People use their fingers to count and then count the number of fingers. All of these operations become much more efficient if you introduce numbers and you get the added benefit of calculating but it’s not an intrinsic part of these operations.

"Mathematics’, like every other field doesn’t have clear boundaries but one can talk about its core nature perhaps. My personal description of Mathematics, is that it is the investigation of systems of abstract objects, like numbers, in search of useful or interesting patterns.

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Reading this is a pleasant diversion from practicing soroban, two digit multiplication and doing my back taxes. My exposure to the real numbers has been limited to a brush or two with calculus. I have just started to read algebra by accident and am only just now learning what abelian means. It is interesting that the various objects have similar rules in this game and can potentially stand in each others stead or mix and match. That is a whole new can of worms. Complex numbers are just a whisper for me on the occasional graphics algorithm.

My initial response was intended to suggest that the “correlation” between math and physics seems to come from the “correlation” between numbers and things that we count (arguably integers). Math appears to give us a lever to “imagine the unimaginable”/extrapolate that suggests that the objects described are not “real” OR at a minimum not provable as “real”. Which is probably close enough for both sides of the conversation between science and math.

Even within math ( my recent reading on Ramanujan has been fascinating. I’m still a Euler fan but Srinivasa would have been fun to hang out with at a party.) the unlikely relationships and patterns between geometry and algebra baffle me. On the other hand, there is much that is approachable as a hobbyist and the fun part of a hobby is that you can take trips down into mazes and caves that you have no time for as a student. As an undergraduate, I did well in my calculus but learned nothing at all. 30 years later I understand it far better but can do almost none of it. (On my list of things to practice again).

Thanks for your thoughts.
-R

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If you ever want to understand this I can write a small primer on when mathematics went from 1d to 2d.
This will help you understand it better and then you can mentally calculate the square root of i for example.

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Let me say first that this subject is not up for grabs. This is not a debate about the nature of numbers, the rules are laid down clearly.

You can have your own version of chess or checkers which you play in your family or your town but if you want to know how chess is actually played in tournaments, if you want to read those games and understand them then you must know the rules as spelled out by FIDE the international chess organization.

Likewise, you can have your own private feelings about numbers but when you come to Mathematics, to understand the work of professional mathematician, the algebra and the theorems it supports, then you must depend on the formal construction - nothing else is relevant and will simply lead you astray. I have seen this happen in practice many times. People have trouble grasping some mathematical idea because they are too attached to their naive intuitions about numbers. It is not an asset.

This thread is not about debating that, it’s a given. It is about understanding and grasping this fact, which as I have carefully explained, does not come easy or naturally to the human mind.

It is not appropriate to stick one’s head in the in the door momentarily and make some half baked foundational attack on the whole on the whole concept of numbers. “What about Godel’s Inconsistency? Eh?” One gains no ground by attempting to undermine the number system. They are a major part of one’s life implicitly or explicitly. They were used to make almost everything we own and eat. One cannot opt out by dismissing numbers as an unsound system. One can try to ignore them, better yet try to understand what one is dealing with, as is the point of this thread.

There are indeed profound foundational questions about the integrity and coherency of the number system, but that’s simply because there are profound foundational questions about every kind of human understanding. It’s limited. We are never completely sure of anything. We can’t even demonstrate for certain that we are not living a dream. To hold up mathematics as if it had a special problem in this regard is, well, a failure to argue in good faith.

But I don’t flinch. I will briefly discuss some of these issues.

I Am The King Of France

*(proof to follow)*

About the most devastating flaw that a mathematical or logical system can have is a Contradiction. It brings down the whole structure.

“If you accept a contradiction you can prove anything

I’ve heard this attributed to Russel, but the proof has been known since the Middle Ages. It’s a set piece and worth knowing. I’ll do this in symbolic notation or it becomes too long.

Notation. We use cap letter to denote statement. P is equivalent to asserting that P is true.

The & operator (AND), P & Q is a compound statement only true if both P and Q are true
The | operator (OR) P || Q only false if both P and Q are false true if one or both are true.
The ! operator (NOT) !P P is not true

Let P = “The sun is yellow” then our contradiction is

  1. P & !P (the sun is yellow & the sun is not yellow)

Let Q = “Adam is the King of France”

Show that P & !P=>Q

2 => !P (since both sides of 1. are true , each is true on its own)
3. => !P | Q (since !P is true, the compound is true)
4. but from #1 we have P
5. But #3 is true and if P=> !P is false, then #3 is true because Q is true
6 P & !P => Q QED

But you all can still call me Adam and treat me like I’m just one of the guys.

If the algebra doesn’t speak to you, Russell was once challenged by a member of the audience to start from 1=2 and prove he was the Pope.

Russell:

The Pope and I are two people.

1 = 2 => 2 = 1

2 = 1 => The Pope and I are one person.

QED

Contradictions are used in Mathematical proofs. A contradiction or a statement which implies a contradiction is FALSE.

An example of Proof By Contradiction

Thm: There is no largest integer

Pf:

H: Let N be the largest integer (the contradiction hypothesis)

=> M = N+1 is also an integer ( basic from the Construction of the Integers CI)

and M>N ( ditto)

and H => Contradiction

=> H is false and there exists no largest integer

QED

A very simple proof of an obvious fact but the principle is an important one.