Why can't numbers be 'used up'?

Question

I was speaking with a young student who has been learning about addition and subtraction (essentially functions, but he doesn't know that yet) with the idea of a 'number machine' and he could not understand how when you put '2' into the machine it's not gone or 'used up', you can get it back and put it into the machine again, unlike a real object which gets turned into something else. Although it seems trivial at first, there are a few interesting implications about his question, why do we talk about abstract ideas as if they are real objects? And from this, is it a property of abstract 'objects' or ideas that there is no situation where they do not 'exist'?

Answer 1

Does a song get 'used up' when we sing it? Does a story get 'used up' when we read it? Does a path get 'used up' when we walk it?

Forgive the computer science analogy here, but all of these things — songs, stories, paths, numbers, and more — are templates or classes that are instantiated when we use them. The concept {2} isn't merely a large box containing every concrete '2' in the universe, which we could slowly use up over time (just as the song "The Sky Is Crying" isn't a set containing every possible rendition of itself). The concept {2} is like a tiny machine in our heads that produces a new '2' any time we need it. When we use a '2' we may be using some other resource (brain power, maybe?) that can itself get used up, but as long as we have that resource we can produce a '2' on command.

Answer 2

Actually, your young student friend may be contemplating an astoundingly subtle notion. Linear logic (invented (or is it discovered?) by Jean-Yves Girard) is a substructural logic that's resource-aware, i.e., premises are "used up" (called "discharged") when used in a proof.

Trivial example, A==>B & A==>C   .=/=>.   A==>B&C because the A premise gets discharged proving either one of B or C. But there's also an "exponential" operator !A that provides for unlimited re-use of A in the usual classical logic way.

So there exist universes of discourse where either idea about the concreteness of "abstract objects" is the prevailing idea. And your young student's notion of resource-awareness is perhaps just channeling Girard. And since he's arguably the world's greatest living proof theorist, I certainly wouldn't go and discourage your young friend's thinking in any way whatsoever.

E d i t
In reply to @Brondahl's comment that perhaps my answer wasn't intended to be serious, let me briefly elaborate why it first crossed my mind that some intuitive notion of resource awareness might have motivated the young student's thinking...

The op's post mentions 'number machine' in a context that suggests he explicitly used this phrase during the discussion with his young student. And machines of just about any kind take concrete input that gets "used up" to produce the machine's output. For example, a bread-making machine takes flour, water, etc as input, from which you get a loaf of baked bread output, but you don't get your flour and water back to be re-used. So as soon as "machine" was mentioned, this resource-awareness notion might very easily have naturally crossed the student's mind.

And in reply to @Brondahl's subsequent comment, " have you not realized that the 'student' in question is a child?", yes, like everyone, I immediately realized that   ( so, do you see, @Brondahl, how, unlike my overall post, that preceding remark actually does possess just the very slightest touch of sarcasm? :)
      My point mentioning linear logic was >>not<< to suggest this child was contemplating any kind of symbolic formalism or its semantics. But rather, that he might have some pretty insightful mathematical intuition, which could serve him very, very well once his education progresses to the point where he can formally develop those insights.
      And his present-day naive expression of those insights should be heartily encouraged, not discouraged. Too many smart young kids get frustrated by pedantic teachers who just shove canonical mathematical gospel down their throats, rigidly "correcting" every original thought that crosses their minds. And that just sours them on mathematics entirely, and they go on to some more "creative" field where they can flourish.

Answer 3

Monkey Brain

Although it seems trivial at first, there are a few interesting implications about his question, why do we talk about abstract ideas as if they are real objects?

Because that is the most efficient way for our monkey brains to process them. We have a tremendous amount of neural hardware for locating and obtaining bananas and the like, and virtually none dedicated to pondering abstract concepts. The fact that we can impose abstract ideas onto our concrete neural circuits is itself a minor miracle (attested by the fact that few other creatures are capable of this feat, so far as we can tell).

And from this, is it a property of abstract 'objects' or ideas that there is no situation where they do not 'exist'?

A physical object "exists" if we can observe it. Can you see it? Can you touch it? Can you hear it? Can you smell it? Can you bounce a rock off it? Can you move it? If there is no way to observe it, then you have no basis to say that it "exists". Of course, our ability to observe is limited, and so the human definition of "existence" is also limited.

Existence

Since we already established that our abstract notions are grounded in physical reality, it stands to reason that "existence" for an abstract concept boils down to: "Can you 'observe' it?" But the abstract realm is not like the physical. We can't bounce photons or rocks off abstract ideas. So for abstract ideas, they basically "exist" if we can hold them in our minds. There are surely abstract concepts that we cannot hold in our minds, because we know our minds are quite limited. And in that sense, we know that for humans, such concepts "do not exist". On the other hand, if we say that "abstract concept" is something that humans invented, and thus, is limited to human existence, then we could just as well say: "Nothing that humans cannot conceive is an abstract concept." In that sense, all abstract concepts must exist, by definition.

Conservation

Fundamentally, the reason that physical objects get "used up" when we "process" them is due to conservation of energy. If you drop a tennis ball into a ball launching machine, the tennis ball does not spontaneously clone itself and remain in your hand while simultaneously dropping into the machine because on a local level, the total energy of an isolated system remains the same. This is just an observed property of the physical universe.

On the other hand, abstract concepts are not composed of "energy" in the physical sense or any related sense. And thus, there is no requirement that the amount of abstract "stuff" is conserved anywhere or at any time. We can easily "clone" an abstract object because doing so is "free". However, we should be careful, because when we talk about humans pondering abstract ideas, what we are really talking about is a physical process representing a virtual one.

When we say: "I picture an apple in my mind. Now I picture two apples in my mind" what has really happened is that some atoms in my brain are in one physical configuration, and then some time later, they are in another physical configuration. If we reduce my brain to a giant panel of light switches, then we see that "copying" the "abstract apple" amounts to simply flipping the switches from one configuration to another. And this is where it gets tricky. My ability to "clone virtual apples" is limited by how many switches I have. This process cannot carry on indefinitely, even though I can cheat and use abstraction to squish the virtual apples into tinier and tinier spaces by using mathematical notation, etc. I can pretend that there are an infinity of virtual apples in my mind, but I can never prove it by demonstrating each one to you.

In this sense, abstract concepts are limited by the rules of physical reality. But because the "atoms" of concepts are merely the physical arrangements of atoms and electrons, concepts operate by a different set of "laws" which are far more flexible than our physical ones. They are not infinitely powerful, but they are much more powerful than physical laws (in terms of what operations and states are allowed).

The number '2' does not get used up when someone uses it in a calculation because the number '2' is just some configuration of matter that can be easily replicated. The only way that '2' would become conserved like a physical tennis ball is if representing '2' in a physical system somehow prevented all other physical systems in the universe from doing the same. That would be an extraordinary set of physical laws which would prevent such a thing, especially given that there is no fixed representation for '2'. One could argue that it would be impossible, because someone could just randomly assert that some other observed configuration of matter does, in fact, represent '2'.

Definitions

A rock is a rock whether there are any humans around to witness it or not. It has an independent physical existence. The number '2', however, requires a human to exist (unless, of course, some other animals have figure out how to count). But by the same token, the number '2' exists anywhere a human says it does. And that's why it would be difficult to impossible for physical laws to constrain where '2's occur in the universe.

Answer 4

I suspect the problem is that the child is imagining a machine with preset number tiles. When it spits out the only tile with a "2", of course it's used up. If you clarify that it has a huge supply of blank tiles and a printer, the problem goes away.

Here's my non-frame-challenge answer:

You could show them. Kids' intro to numbers starts with concrete objects and then generalizes into the abstract -- a 2 is just 2 of whatever. We can step back into the concrete world. Make a pile of 2 raisins, then place -- plop, plop -- some raisins in another pile. The new pile also has 2 raisins. Clearly the 2 wasn't used up by the first pile. Obviously we can put have any number of raisins in any piles we want.

We can also see where used-up could apply. If we have number-signs then of course we use up the "2" on the first pile of raisins. We can't label the second pile with a "2" sign, But it still has 2 raisins. We just have to count them each time, or write it on a blank sheet.

Answer 5

... he could not understand how when you put '2' into the machine you it's not gone, you can get it back and put it into the machine again.

One potential idea to think about here is to "design" some number machines with him!

So imagine the base case - let's take an example of a number "machine" which is actually just a number car wash. It takes your number "2", makes it all shiny and clean, and returns the same number 2 that has just had a bit of a clean.

(this is the identity function - it's a function from N to N that associates each number in the domain with an identical number in the codomain)

Now, we want to consider a function that is in some way transformative. Well, we can't straightaway design machines that work with numbers themselves, but we can design machines that work with representations of numbers. So let's start with a very simple "incrementer": you have a slot where you put in some number of marbles, and when you press a button, it pops out both the marbles you put in and one additional marble into a tray.

Hang on a moment, the student says - this isn't a number machine, this is a marble machine! And that's quite correct; the defined "domain" of our machine is the sets of marbles and the "range" is also the sets of marbles. Our machine doesn't work for e.g. Skittles, because if our machine takes some Skittles as input, it'll either not work at all, or maybe if we're lucky it'll give us our Skittles back plus a marble.

But what if we came up with a super high tech upgrade to our marble machine? Let's suppose we have a machine that can tell what kinds of things have been put into it, and by scanning the things you put into the slot, it says "if all of these things are the same, then we will copy one of the things put in and add it to the group we send out". This way, your machine isn't just a marble machine, because if you put a bunch of Skittles in, you would get an extra Skittle instead!

Okay, so what we've got is a copy machine; we're adding one of the same kind of thing. But the key thing is that it's a kind of copy machine that works for anything - that's the key thing that makes it a "plus one" machine. Similarly, you can also imagine a delete machine that works the same way as a "minus one" machine.

So now we can start doing silly stuff like connecting the tray of one machine to the slot of another machine! So imagine you put two apples in one slot, it goes through the copy machine, then those two apples plus the new copy apple go straight into another copy machine, and in the ray of that machine you get four apples. This is a Plus Two machine. And so on and so on.

The thing that makes the transformations like "adding 4" and "subtracting 2" special here is that the numbers aren't the actual things that are going into the machine. Really, it's more accurate to say that "the number" of things being fed in to the machine is a quality of the set of things being put in - specifically, its cardinality. But our special number machine is something that works whatever kinds of things are in the set, and that concept of cardinality is something that is relevant to any set of things, including both things that go into the machine and things that come out of the machine.

And now it should be clear that people can do things that are very like specific particular instances of what this special machine does in general! If I want to take a set with two things in it and get a set with four things in it, I can find another one of those things, then find another of those things, and clump the whole set together. And I can do this in my imagination - often a whole lot easier and with way weirder and cooler things than I can in person!

Answer 6

Short Answers

[W]hy do we talk about abstract ideas as if they are real objects? And from this, is it a property of abstract 'objects' or ideas that there is no situation where they do not 'exist'?

1. Human beings' reason is heavily suffused with the use of conceptual metaphor, figurative metaphor, analogy, and argument by analogy especially in abstract domains such as a mathematics.
2. The process of abstraction is essentially boundless, as signs and rules occurs as a result of a faculty of generative grammar of the brain (at least according to a naturalized epistemology).

Long Answer

Language, Similarity, and Concept

The human brain is unique in the biological world, because while our Great Ape cousins are capable of using signs, like Koko the gorilla, a human child acquires grammar-based language and almost immediately begins putting it to use. What makes a language different than a mere sign system used in the the rest of the animal kingdom is a grammar. Grammars are often compared, at least in computer science, based on their expressivity. For instance, some cultures have but 3 words for numbers: none, one, and many. Obviously in the global mathematics community, our vocabulary for discourse about numbers is very expressive including ideas like transfinite numbers and diagonal proof. The child you are referring to is on a journey from subitizing, pairing, and counting to conjectures, proofs, theories, and maybe someday mathematical logic.

Two books might be of interest to you. One is Where Mathematics Comes From, which is two cognitive scientists' philosophical conjecture based on other views that are offered in the philosophy of language about the interface between thought and language. The other is Surfaces and Essences: Analogy as the Fuel and Fire of Thinking which comes from a computer scientist and psychologist duo and seeks to explore how reason generally occurs. From the book:

What we mean by [analogies and concepts playing a central role in thinking] is that each concept in our mind owes its existence to a long succession of analogies made unconsciously over many years, initially giving birth to the concept and continuing to enrich it over the course of our lifetime.

The Function Machine

Thus, when you ask 'why does a child think this way about consuming', the only clear and resounding answer is that when one uses machines in the literal sense, let's say you're producing a cake batter in a mixer, you use up your ingredients. In analogy, we extend properties from one thing to another, sometimes erroneously. For convenience, let's call the literal domain the source domain, and the mathematical abstraction the target domain.

Source domain: A mixing machine is a tool that takes basic inputs called ingredients like eggs, flours, and milk and through mixing changes them into an output called batter which we store in a container called a bowl.
Target domain: A function machine is an abstract tool that takes basic inputs called numbers like integers, whole numbers, and real numbers and through arithmetic operations changes them into an output called range values which we store in a set.

Do you see the analogy? We have the same language at play regarding inputs, outputs, processing, and storage. The major difference is one example of change is physical where the other is mental. Thus, the information processing cycle is a model that describes (however perfect or imperfect) what goes on in the world. In fact, that's a central thesis in a branch of philosophical theories called physical computation (SEP).

False Analogies

But alas, in some way, all analogies are false insofar as they aren't true equivalences. They merely approximate explanations. And as we grow more sophisticated as thinkers, we become better as using logic to see where they fall short. Given that real world machines and functional machines have such structural similarities, it's very reasonable to confuse the two concepts and impute all of the properties of the source domain to the target domain. But as you know, mathematical objects are not physical objects (though there's a long standing defense and great deal of popularity to Platonic thinking that mathematical constructivists take issue with). But that's a different Q&A entirely. See What does mathematical constructivism gain us philosophically? (PhilSE) for an introduction to that topic.

Answer 7

Free new instances of the same class

Perhaps the proper analogy to use here is to note that each invocation of the number 2 is a 'separate instance' of the general concept of number 2 - you may obtain 2 in different ways (2, 1+1, 4/2, etc), and all those 2s (note the plural) are equal but you can treat them as separate instances if you want (and in some scenarios in e.g. software engineering, you might have to).

And these numbers don't get 'used up' because they come with the useful property that we can freely summon new 2s or clone existing numbers, so if we have some result of a computation we can at the same time transform (an instance of) it and also keep (an instance of) it unchanged.

Answer 8

Numbers are names

Numbers are names and functions are not machines. Your name, say "Roger", does not get used up when your friends refer to you. If you have two apples in a basket, you can give it the name "two" if you are interested in how many things there are. You can also give it names like "red" or "food", if it suits you. Numbers are those names that you can add and multiply (among other things). When you use names without having the actual things at hand you are abstracting. You can say "Kim" without ever having known anyone named Kim and you can say "1983423" without having counted that far. Abstraction lets us keep our "eyes on the ball" and focus on the important tings without geting tangled up in details.

Functions are not machines that you shove stuff into. Functions are special relations between things or names. What makes functions special is that they always work, that is, if it can fail, it is not a function. When kids are standing in a circle, "to the left of you" is a function. You can look to your left and find Sally there. So, "to the left of" "Roger" = "Sally. Neither Roger or Sally need to move or get used up. Another function is "Mother". Since every human (alive today) has a mother, it is a valid function. In this case "Mother" "Roger" = "Emma". This reads as "the Mother of Roger is Emma". You can also use the Mother-function analogy to talk about functions that are not easy to compute. We might know that something is a function without knowing what the value of the functions is. It is valid to say "Mother" Steve = ? since we know Steve has a mother, even if we don't know who that is.

Answer 9

This discussion is missing the concept of units, which I think the child is intuiting (correctly).

You give Alice 2 apples. She had 3 apples. Now she has 2+3=5. Do you have the original 2 any more? No. That pair got "used up" and is now part of a new group of 5. You can't give those 2 apples to Bob unless you go get them back from Alice first (destroying her "5").

Numbers count things, and each equation is a story, and the question "does the 2 get used up" depends on the story.

Answer 10

I think that's a really interesting point that your student has a doubt in. I believe the issue is that when we do calculations, the numbers and related concepts are something we come up with our brains to describe it. It needs to be in no relation to the world. In our minds we can go and calculate the sums and other quantities which may have a physical relation but ultimately the number is simply an idea which exists in our mind. So, we could do anything we want with it because it exists in there only.

This even extends to any quantitative means of measurement. For example, length , time etc. All of those are fundamental only in the sense of our understanding, if we perhaps met an advanced alien race for instance, they would have some deeper ideas which are more fundamental. The ultimate conclusion is that only through the numbers mean nothing in the realm of application until the final step where the result we did using the number is translated into a English or a concept the mind can perceive.

Answer 11

As others have remarked: Because the mathematical "2" is not a physical object but a thought, an idea, a concept, a property. Existing purely within our minds, it has almost no physical reality, so that the conservation laws governing the physical world do no apply. (We do need food and oxygen for the physical substrate our mind is running on, and our time is limited, so that at the end of the day there is a limit to the amount of "2"s we can imagine, but then we can use mathematical abstractions and transcend that last tether to the material.)

So why is it that we are able to imagine multiple "2"s? It is because our brains have evolved to perform planning. We can think ahead and act strategically and adapt our behavior to achieve desirable outcomes: Follow a plan. There is a yummy fruit hanging in the tree and our ancestors pondered what's the best way to obtain it: Use a stick, climb the tree, wait until it falls down. That's three times the same fruit. We duplicate the fruit in our heads. Or, more correctly, we learned from Magritte that we duplicate our mental image of that fruit. Simple animals cannot do that: They can only react to concrete stimuli: One concrete apple. Law of conservation.

The ability to manipulate (and, in that process, "multiply") symbols in our minds is an outflow of our ability to make plans, which requires abstraction and symbolic thinking.

On top of that, abstraction is a devilish thing. Nothing keeps you from applying abstraction to abstractions — in fact, that is what higher mathematics is all about. You can remind your student that she can easily imagine two concrete objects, say, two existing apples. She can take them away from a heap, add them to another heap, eat them. She can imagine them twice. A million times, if she wants. Indeed, anything is arbitrarily often available in her mind. Multiple moms. Multiple selves (but perhaps don't spook her).

So our mind manipulates an internal symbol of two apples. Now we can abstract from the concrete apples and think of two generic apples. Abstract again: Two things. Abstract again: Two. The common property. Now it's not amazing any longer that we can produce twos to our heart's desire: We could do that even with real, existing apples. It should be even easier with something that doesn't correspond to anything in the physical world any longer.

Answer 12

A function is a process, algorithm, or explicit mapping. Rather than talk about a machine churning up the input number to produce the output number, I would instead compare the act of applying a function to saying a number to someone, having them carry out a process using that number, and then telling you the result of the process.

Suppose Alice and Bob are playing a game where Bob has a function f in mind and Alice is tasked with figuring out what the function is, but all she can do is ask Bob what f(x) is for any particular x. The conversation may go like this:

ALICE: What's f(2)?
BOB: It's 7.
ALICE: Okay, how about 5?
BOB: It gets mapped to 18.
ALICE: What about 2?
BOB: It still gets mapped to 7, the function is deterministic.

The fact that Alice asked about 2 already and that Bob carried out the process to figure out f(2) = 7 does not preclude Alice from asking again, nor Bob from carrying out the process again.

Answer 13

To your question, we treat abstract ideas as real objects because many people can comprehend a real object and not the abstract idea. Or rather, we try to present the idea in a form that the listener can comprehend with the challenge of choosing a single form that will work with most listeners.

Toward your student, find something they are passionate about or something that is familiar to them. Whether we use a crayon, a skittles candy, guitar pick, coin, or even a slice of pie, the analogy will be the same. We can have multiple things that look nearly identical and have the same base value but consuming any one of them does not use up the others.

Answer 14

The closest answer to this question that has been posted already, is that of Mankka, which states that numbers are names. However, this does not take it far enough. My answer is that the existence of numbers is independent of their representation.

Lets start with a simpler question: What are natural numbers? Well, the set of natural numbers is nothing more or less than an abstract concept of something that exposes a certain behavior. I.e. you start by saying:

1. "0" is a natural number

2. For every natural number "x" there is a natural number "x+1"

3. For natural numbers "x" and "y", "x > y" is true if and only if "x = y+1" or "x > y+1"

4. "x > x" is false for all natural numbers "x"

5. "x + 0 = x" for all natural numbers "x"

6. "x + y+1 = x+1 + y" for all natural numbers "x" and "y"

And that's it. You postulate that something called natural numbers exist, you require them to behave according to some rules, and then you work with them. This does not call natural numbers into existence, though, it merely gives them names! In fact, you can write down other definitions for natural numbers, but as long as you can prove that the set of rules above implies the rules in the alternate definition, and that the alternate definitions' rules imply the rules above, both are talking about the same thing. And the same goes for the number "2": Assuming the usual interpretation of "2", you have that

2 = 0 + 2 = 0 + (1+1) = (0+1) + 1 = (0+1)+1

which precisely defines the position of "2" in the sequence of natural numbers, and "2" is simply another name for the natural number "(0+1)+1".

As such, numbers do not get used up when they are "put into" something, because they exist outside of space and time, and we only ever use different forms of names (=representations) to work with them. And when you do something with a name, you are not changing the named thing in any way.

Answer 15

Our capacity of 2s is very large

When you use 2, you only use a tiny bit of our capacity to make 2s.

Now, some will say this is ridiculous, it is an abstract thing, just a name. But information itself is concrete.

We live in a universe of increasing entropy. Entropy has been described as chaos or randomness, but it could better be described as a process of accumulating information trash.

Every bit of information processed in this universe leaves a kind of residue or noise. In theory, this residue can be used to reverse the processing of the information, but this reversal itself would (in practice) produce even more information residue. This accumulation of noise is what we call entropy.

The concept of 2 -- using it -- is a tiny bit of information processing. The total supply of 2 in the universe is insanely large. Every time you use the concept of 2, you are increasing the entropy of the system, and there is no going back.

We live in a universe with extremely low entropy. In an extremely high entropy universe, it is very smooth.

Counting is concrete

Counting, for example, requires there be distinct things. In a high entropy universe, things are smeared out so much that finding 2 of something (or even 1) is impossible, let alone a system that can model 1 or 2 internally after interacting with it.

The current projection is that the universe ends in what is known as "heat death" and proton decay.

In the far, far future, every proton has decayed. The universe is full of stretched out photons and isolated electron clouds. The electrons, with next to nothing to interact with, form probability clouds that become beyond astronomical in size, and "where an electron is" becomes nearly meaningless. As electrons and photons have no hair, which part of the electron field is one electron and which is another is also meaningless.

In short, we will run out of 2. And the process of using a 2 works towards this.

The time lines we are talking about here are extremely long. As in, if you tried to draw the timeline on a graph the size of the observable universe, the total time since the big bang to today would be much smaller than a single atom on that timeline.

TL;DR

The universe has an extremely large capacity of 2s. We can make an instance of the concept 2 using other parts of the universe, so we won't run out very soon. But we believe we will eventually run out of 2s.

Answer 16

Definition of "used up" from a philosophical perspective:

According to the Principle of Conservation of Mass, no matter disappears, as in "using up". Matter would just change. So, when does something get "used up", if matter does not disappear, but just changes?

"Used up" is just a subjective judgement.

It is reason that defines the boundaries between what is/is not "used up", subjectively, learning by experience. So, when you eat an apple, it becomes "used up" after it get transformed into food bolus. A paper becomes "used up" when a surface around drawn lines occupies certain percentage of the surface. An ink cartridge becomes "used up" when a percentage of the ink is transferred to the exterior, etc.

Ideas (e.g. numbers) don't get "used up" in such sense, because a) they are not made of matter, but are rational concepts, and b) they can remain "pure" as ideas and don't evolve in our brain.

Nevertheless, some ideas might evolve (e.g. the memory of a face)... and so, they might get "used up" in the same exact sense of change and material objects being "used up".