What do we mean by the term "Number of things"?

Question

I am reading the book "The Number-System of Algebra (2nd edition)." I have some problems with the the first article: "Number".

The author has confined the concept of number of things to the groups which have all distinct elements, that is the number of letters in a group having elements A,B,C is 3 iff A,B,C all are distinct.

What is the definitions of the term number of things in general English?

My understanding about the term number of things is that when we talking of some concrete things then we are interested in knowing how many concrete things(tokens) are there. We do not bother whether the concrete things under consideration have similar properties or not.

When the things under consideration are "abstract objects" then we are only interested in knowing how many different types of "abstract things" are there. For example consider a child learning English alphabets. The student writes the letter "A" 10 times, the letter "B" 3 times and the letter "C" 2 times. the teacher asks the student:
"How many alphabets you have learnt to write?"
The child will reply:
"I have learnt to write three English letters, namely "A","B" and "C"."
The child has actually written 10+3+2=15 letters but it is understood that the teacher meant to ask "how many types of letters".

Mr.Fines book is quite old. I want to read some latest literature for understanding the term The Number of things.

Which field of study deals with this term(Number of things)? Dose Modern Math or Modern Philosophy deals with this term? Which subject I should read for the formal study of this term. Does modern set theory deals with this term?
Could you guys tell me about some modern book which formalizes this term. I have downloaded the book "Recursive number theory (1957)" but this seems to be old.

Answer 1

The book is a very old one : 2nd ed 1903; 1st ed 1890.

As you can see from footnote page 131, Cantor and Dedekind are mentioned as "interesting contributions to the literature of the subject" ...

Thus, you cannot expect that the concepts introduced at the beginning without definition, used as primitive in order to "elucidate" the following treatment, can be exactly translated into modern (i.e.post-1930) set-theoretical notions.

I think that :

group must mean a finite collection of objects (things)

and that :

number of things in a group is "clearly" (from the discussion) the equivalent of modern cardinality (restricted to finite collections) and it is called a "property" of the collection (group).

My interpretation is that things are "individual", concrete or abstract (if any). Of course, it is easy to think to them as concrete objects, like peebles in a pocket or soldier in a platoon.

A platoon is a group of soldiers and the number of things in the platoon is the number of individual soldier forming it.

This interpretation makes sense also with regards to the ensuing definition of addition (see CoolHandLouis's answer).

Please, note that here group has the "generic" meaning of collection or aggregate; it has nothing to do with the technical term "group" of group theory.

When we "abstract" from the "characters" of the individual things (i.e.form their individual properties, like colour, size, shape for a colelction of balls) and from the order of the objects in the collection (it is the same for the "modern" set concept: { A,B,C } is "the same" set as { C,B,A }) what we obtain is the "number" of the things in the group (the number of the members of the collection).

Remember that Cantor's original notation for representing the Cardinal number of the set A was a "double overbar" over A :

the symbol for a set annotated with a single overbar over A indicated A stripped of any structure besides order, hence it represented the order type of the set. A double overbar over A then indicated stripping the order from the set and thus indicated the cardinal number of the set.

Answer 2

Preface

I provided two answers to this question:

• The other answer is the better answer and is my primary answer. It suggests Mr. Fine is referring to naive set theory.

• I provided this answer because the OP insisted on thinking of {A,A,A} as containing "three distinct elements" and posted a bounty. There was absolutely no convincing OP otherwise, so why not just agree and get the bounty? :)

  The two answers actually complement each other since they show how one can describe the same mathematical phenomena by changing axioms, definitions, and rules in different places. You say TOE MAY TOE I say TOE MAH TOE. As it turns out, this answer contains a cute "mathematical proof" that Mr. Fine thought {A, A, A} represents three distinct elements". But please do feel free to read a tongue-in-cheek attitude in this answer.

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Anupam,

You are right Mr. Fine considers {A, A, A} = 3.

I'm submitting another answer because I figured this out, but wanted to leave my old answer for the sake of history. You are right! Henry Burchard Fine meant three concrete things so {A, A, A} is counted as three. His statement cannot be a mistake because it's his primary premise in substantiating all of his numerical arithmetic - the basis of his entire book - starting with addition:

Addition: If two or more groups of things be brought together so as to form a single group the numeral symbol of this group is called the sum of the numbers of the separate groups.

If the sum be s and the numbers of the separate groups a b c etc respectively the relation between them is symbolically expressed by the equation s = a + b + c + etc where the sum group is supposed to be formed by joining the second group to which b belongs to the first the third group to which c belongs to the resulting group and so on

The operation of finding s when a b c etc are known is addition. Addition is abbreviated counting.

6 Addition If two or more groups of things be brought together so as to form a single group the numeral symbol of this group is called the sum of the numbers of the separate groups If the sum be s and the numbers of the separate groups a b c etc respectively the relation between them is symbolically expressed by the equation s a b c+ etc where the sum group is supposed to be formed by joining the second group to which b belongs to the first the third group to which c belongs to the resulting group and so on The operation of finding s when a b c etc are known is addition Addition is abbreviated counting

• Given a, b, c are "groups/sets",

• If two or more groups of things be brought together so as to form a single group...
  Let d = a U b U c

• ...the numeral symbol of this group is called the sum of the numbers of the separate groups)
  Sum(d) = Sum(a) + Sum(b) + Sum(c)

• Now define the groups/sets as follows:

  • a = {A}
  • b = {A}
  • c = {A}

• Sum(d) = Sum(a) + Sum(b) + Sum(c) = 1 + 1 + 1 = 3

• d = a U b U c

• Therefore, Mr. Fine's "union operator" must be creating d = {A, A, A} and sum({A, A, A}) = 3.

• If Mr. Fine's "union operator" was normal set notation, then d = {A} and there is no way one could obtain "3" from that.

Therefore, Mr. Fine considers {A, A, A} = 3.

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This is the case when A represents distinct concrete objects, like 3 coins in a pocket.

Answer 3

The work that first comes to mind is Edmund Husserl's Philosophy of Arithmetic. He addresses in some detail the obvious difficulty with number: that to count the things counted must be both different (so there may be more than one) and the same (you are counting certain things). When I say "three apples" they are all the same in one sense (they're apples) and they are all different in another (there are three of them, distinguished by their spatial relationship if nothing else)

There is simultaneous "multiplicity" and "unity". This leads to the question "the same in what way, and different in what way".

The thing I remember the most from this book is discussion of difference and distinguishing. It's something worth talking about. There are two terms that can be contrasted, "different", "distinguished".

• To distinguish between two things we must make a judgement
• Different is a nescessary but not sufficient condition for things to be distinguished

In mathematics everything that is different is distinguished and one considers a totality of distinct things. This avoids the tricky part: human judgement.

This judgement is often easy for us. It is clear that we perceive many things as distinct and that the world "crystallises" into objects. Although this perception is not always all that is needed for distinguishing between things, in most day-to-day situations it is enough. It is only in edge cases where we need to go beyond our the appearance of objects separated in space, and use some other mode of judgement.

The ability to distinguish between things is the main topic of the scientific field of psychophysics, which really got going around the 1890's and continues to this day. There have been many philosophical writings about it this human capacity too, in fact I'm of the opinion that it is the main question of philosophy (others may not agree).

To answer your question directly: mathematics excludes human judgement, so when constructing a formal system we must start after judgement has been made - we do it by assuming that its objects are all distinguishable from each other. If objects in mathematics are not distinguishable they are taken to be the same. This is not true of real things, which can be different but not distinguished.

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Note: The details of how arithmetic becomes abstracted from human judgements is covered in the remainder of Husserl's book. I'm not really capable of articulating it here. I think there might be some problems with it in light of recent scientific research "numerousity". I'm not sure yet.

Answer 4

Preface

I provided two answers to this question:

• This answer is the better answer and it suggests Mr. Fine is referring to naive set theory. Also, there is no great attempt at rigour here, and Mr. Fine simply jumps forward to his topic of interest. This answer is my primary answer.

• I provided another answer in this same thread because the OP insisted on thinking of {A,A,A} as containing "three distinct elements" and posted a bounty. There was absolutely no convincing OP otherwise, so why not just agree and get the bounty? :)

  The two answers actually complement each other since they show how one can describe the same mathematical phenomena by changing axioms, definitions, and rules in different places. You say TOE MAY TOE I say TOE MAH TOE. As it turns out, the other answer contains a cute "mathematical proof" that Mr. Fine thought {A, A, A} represents three distinct elements. It may be interesting to see how I defended such a proposition.

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1. The Book is Referencing Naive Set Theory

The following Google Books link is easier to reference: The Number-system of Algebra: Treated Theoretically and Historically" (Henry Burchard Fine, Copyright 1890, Published 1907). The following is the excerpt in question from this 1907 book:

I. THE POSITIVE INTEGER AND THE LAWS WHICH REGULATE THE ADDITION AND MULTIPLICATION OF POSITIVE INTEGERS

1 Number. We say of certain distinct things that they form a group (by group we mean finite group that is one which cannot be brought into one to one correspondence 2 with any part of itself) when we make them collectively a single object of our attention.

The number of things in a group is that property of the group which remains unchanged during every change in the group which does not destroy the separateness of the things from one another or their common separateness from all other things.

Such changes may be changes in the characteristics of the things or in their arrangement within the group. Again changes of arrangement may be changes either in the order of the things or in the manner in which they are associated with one another in smaller groups.

We may therefore say: The number of things in any group of distinct things is independent of the characters of these things of the order in which they may be arranged in the group and of the manner in which they may be associated with one another in smaller groups.

2 Numerical Equality. The number of things in any two groups of distinct things is the same when for each thing in the first group there is one in the second and reciprocally for each thing in the second group one in the first. Thus the number of letters in the two groups A, B, C; D, E, F, is the same... [Mr. Fine continues to talk about 1-to-1 correspondance - CoolHandLouis]...

It is clear to anyone who takes a beginning-level "Set Theory 101" class that this book is describing the foundation of set theory. We can confidently say that Mr. Fine's references to a "group" is exactly and precisely what is now known as a "set", and to "elements" when he was describing "distinct things". (As an aside, this entire post actually refers to what is called "Naive Set Theory", but that is inconsequential for this question/answer.)

Given that Mr. Fine is referring to Set Theory, and his book was written in 1907, my first suggestion is that you forget about Mr. Fine completely and google for some good references for beginner "set theory" and also look at some of the short videos on the same subject.

Mr. Fine's footnote "By group we mean finite group that is one which cannot be brought into one to one correspondence with any part of itself" is very strong evidence he's talking about (naive) set theory. He's obviously avoiding infinite sets, and based on the history of Set Theory, that may have been for political reasons. There's no reason for him to be contentious at that point in his career, and every reason to play it safe, especially with this book.

But that's a meta-answer. Here's a real answer:

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2. Answer to Question - Intro

First lets standardize the rest of this post's language to the 21st century: A set is a collection of distinct elements. So let's not talk of "things" or "groups" anymore. And it matters not if they are concrete or abstract, real or imagined.

Changing the names for these terms does not in any way change any of the issues you are encountering. The new words refer to exactly the same thing Mr. Fine was saying. It's all a matter of definition, and I'll define everything as we go to show you the difference that is causing confusion.

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3. How You Are Looking At "Distinct" and "Counting"

First, in one way, you are right. Within your own personal understanding/belief-system/definitions of "distinct", "collection", "set of things" and "group", and how one deals with them, you are "concluding" that "you are right". And neither I nor any mathematician can argue against your "right-ness" in this sense. Based on your definitions and methods of thinking, you are absolutely right. But that's just a start; That doesn't solve the confusion.

Let's make-up/invent a system in which you are "right". (Remember that we could just as well say "groups" and "things" but I'm standardizing to "sets" and "elements". The words used don't make any difference as long as we define them.)

Non-Standard Set Theory Rules According to Original Poster

• A set is a collection of elements.
• Each element is represented by one or more symbols (alphanumeric).
• The size of the set is the total number of elements.
• OP's Definition of Distinct: Each element is considered "distinct" if it appears in a different position, so {A, A} contains two distinct elements because they are in different positions (position one and position two).

Question: How many elements are there in {A, A, A} according to the above non-standard rules by Original Poster? Answer: 3.

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4. How Math Set Theory (Mr. Fine's Book) Defines "Distinct" and "Counting"

Now let's consider this more from the standard mathematical definition.

Standard Mathematical Set Theory Rules

• A set is a collection of distinct elements.
• Each element is represented by one or more symbols.
• The size of a set is the total number of elements.
• Set Theory Definition of Distinct: Each element is considered "distinct" if it can be determined to be different than all other elements. When represented by letters and words, the only concern for distinctness is whether or not elements have different names. In written math, distinct = different names.

For the purpose of this answer, something named the same is not distinct - it refers to the same thing. So {A, A} is like saying, {India, India}. It's only referring to one country, not two countries. It's referring to the same country two times. So which is the count? The one country, or the two times it's mentioned? In set theory, it's the former.

"But why?" you might ask. In a way, you can think of this as completely arbitrary. "It's by definition." (But it's that way for a good reason; it makes a lot of other things in set theory work out well, but that's beyond this discussion). So you just have to accept it, just like "we have to accept that you are right with your definition".

Question: How many distinct countries are there in {France, France, France, France, India, India, India, Brazil, Brazil}? Answer: 3 because the set is just referring to three distinct places = {France, India, Brazil}.

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5. Coins in Your Pocket

It is for this reason and the sake of simplicity that we simply add another rule to Set Theory:

• No duplicates are allowed in sets.

Why? Because a set is kind of like a "bag of things" (concrete or abstract). For example, let's consider four coins in your left pocket on Monday. Let's say we don't know what they are. So we name them C1, C2, C3, C4.

• Monday_InLeftPocket_AllCoins = {C1, C2, C3, C4}

Given this idea, it makes no sense to refer to this as {C1, C1, C1, C2, C3, C4}. Why refer to the first coin three times? It's already in your pocket. It only needs to be referred to once. Now let's assign some attributes to the coins:

• C1 = Type=Penny; FaceValue=0.01; Date=1999; Weight = 2.4993399494 g; Condition=Mint
• C2 = Type=Penny; FaceValue=0.01; Date=1999; Weight = 2.4990044384 g; Condition=Good
• C3 = Type=Nickle; FaceValue=0.05; Date=2002; Weight = 5.0002292833 g; Condition=Very Good
• C4 = Type=Nickle; FaceValue=0.05; Date=2003; Weight = 5.0010022229 g; Condition=Very Good

Now that we know two of them are pennies, the set of coins in your pocket is still the same:

• Monday_InLeftPocket_AllCoins = {C1, C2, C3, C4}

But now we can ask about how many different (distinct) types of coins are in your pocket:

• Monday_InLeftPocket_TypesOfCoins = {Penny, Nickle}

Let's move coins C2, C3, and C4 to your right pocket on Tuesday. What's in your pockets on Wednesday?

• Wednesday_InLeftPocket_AllCoins = {C1}

• Wednesday_InLeftPocket_TypesOfCoins = {Penny}

• Wednesday_InRightPocket_AllCoins = {C2, C3, C4}

• Wednesday_InRightPocket_TypesOfCoins = {Penny, Nickle}

Answer 5

Q1: Since $A$ and $A$ are not distinct, only $A$ and $B$ are distinct (unless you are rabulistic and distinguish "the first blob of ink forming an $A$" from "the second blob of ink forming an $A$", but that makes it impossible to mention properly any of these $A$s as the concrete letter (blob of ink) $A$ used to mention a specific letter (blob of ink) $A$ is automatically different from that blob of ink, contrary to intent. In all these cases we speak of the "idea" of $A$, i.e. any instance of "$A$" in the text refers to the same object, which itself is to be thought outside the text (to make it possible in the first place to use "$A$" to talk about $A$). Only in this sense $A=A$ (for as concrete blobs of inks on the paper they have differnt positions, making them different) and the two $A$s in "$A,B,A$" lack distinctness. Your group is thus the same as that having elements $A,B$ (or $B,A$ if you like), i.e. the number is $2$.

Q2: They are still not identical as objects. E.g. You can put on the first and put the second into your cabinet while hot ironing the third; you would shurely notice iit if you were in fact hot-ironing the same shirt as the one you are wearing. The shirts are indiustinguishable by the property "colour" (as they were before that already indistinguishable for instance by the property "size", I assume), but they are still distinguishable by the property "spacial position". Intriguingly, this leaves us with the problem that we run into difficulties how to identify the shirts of today with those of yesterday. One has to think quite a while what "distinct" (as opposed perhas to "distinguishable") and "same thing" mean.

Q3: Distinctness of elements (which may allow identically coloured shirts) is essential, as you do not want to count the same object again (doing so would turn you into a rich man with just a single coin in your pocket). A totally(?) different approach is to define "number" as the equivalence class of sets (and it seems that Fine's "group" is what we would call "set" today) under "equinumerability" (i.e. existence of a bijection between the sets). This way the concept of 2 or Two-ness corresponds to (or in fact is) the class of all sets $X$ such that there exists a bijection form $X$ to any specific set of (what we call) two elements, such as $\{\emptyset,\{\emptyset\}\}$. If you have a horror about (proper) classes, one may notice that each such equivalence class contains a special "simple" set, an ordinal (at least in the finite case, and in general under the assumption of the axiom of choice).

Answer 6

"Number of things" in general English : There isn't enough information in the term alone to give one answer.

The problem is the term "things". In general English this would refer to some arrangement already defined, for example number of items os the same colour or number of eggs in a box, or number of digit "3" there are in a phone number.

Without that, the meaning of "number of things" is manyfold - it's the number of objects in a container of any kind/size, classed by any method you care to imagine.

Answer 7

I suggest you to compare the Fine's definition with the following discussion, from RL Goodstein, Recursive number theory (1957) :

The question 'What is the nature of a mathematical entity?' is one which has interested thinkers for over two thousand years and has proved to be very difficult to answer. Even the first and foremost of these entities, the natural number, has the elusiveness of a will-of-the-wisp when wc try to define it.

One of the sources of the difficulty in saying what numbers are is that there is nothing to which we can point in the world around us when we are looking for a definition of number. The number seven, for instance, is not any particular collection of seven objects, since if it were, then no other collection could be said to have seven members; for if we identify the property of being seven with the property of being a particular collection, then being seven is a property which no other collection can have. A more reasonable attempt at defining the number seven would be to say that the property of being seven is the property which all collections of seven objects have in common. The difficulty about this definition, however, is to say just what it is that all collections of seven objects really do have in common (even if we pretend that we can ever become acquainted with all collections of seven objects). Certainly the number of a collection is not a property of it in the sense that the colour of a door is a property of the door, for we can change the colour of a door but we cannot change the number of a collection without changing the collection itself. It makes perfectly good sense to say that a door which was formerly red, and is now green, is the same door, but it is nonsense to say of some collection of seven beads that it is the same collection as a collection of eight beads. If the number of a collection is a property of a collection then it is a defining property of the collection, an essential characteristic.

This, however, brings us no nearer to an answer to our question 'What is it that all collections of seven objects have in common?' A good way of making progress with a question of this kind is to ask ourselves 'How do we know that a collection has seven members?' because the answer to this question should certainly bring to light something which collections of seven objects share in common. An obvious answer is that we find out the number of a collection by counting the collection but this answer does not seem to help us because, when we count a collection, we appear to do no more than 'label' each member of the collection with a number. (Think of a line of soldiers numbering off.) It clearly does not provide a definition of number to say that number is a property of a collection which is found by assigning numbers to the members of the collection.

To label each member of a collection with a number, as we seem to do in counting, is in effect to set up a correspondence between the members of two collections, the objects to be counted and the natural numbers. In counting, for example, a collection of seven objects, we set up a correspondence between the objects counted and the numbers from one to seven. Each object is assigned a unique number and each number (from one to seven) is assigned to some object of the collection. If we say that two collections are similar when each has a unique associate in the other, then counting a collection may be said to determine a collection of numbers similar to the collection counted.

The weakness in the definition lies in this notion of correspondence. How do we know when two elements correspond? The cups and saucers in a collection of cups standing in their saucers have an obvious correspondence, but what is the correspondence between, say, the planets and the Muses? It is no use saying that even if there is no patent correspondence between the planets and the Muses, we can easily establish one, for how do we know this, and, what is more important, what sort of correspondence do we allow? In defining number in terms of similarity we have merely replaced the elusive concept of number by the equally elusive concept of correspondence.

Some mathematicians have attempted to escape the difficulty in defining numbers, by identifying numbers with numerals. The number one is identified with the numeral 1, the number two with the numeral 11, the number three with 111, and so on. But this attempt fails as soon as one perceives that the properties of numerals are not the properties of numbers. Numerals may be blue or red, printed or handwritten, lost and found, but it makes no sense to ascribe these properties to numbers, and, conversely, numbers may be even or odd, prime or composite but these are not properties of numerals.

The antithesis of "number" and "numeral" is one which is common in language, and perhaps its most familiar instance is to be found in the pair of terms "proposition" and "sentence". The sentence is some physical representation of the proposition, but cannot be identified with the proposition since different sentences (in different languages, for instance) may express the same proposition. [see types and tokens]

The game of chess, as has often been observed, affords an excellent parallel with mathematics (or, for that matter, with language itself). To the numerals correspond the chess pieces, and to the operations of arithmetic, the moves of the game.

Here at last we find the answer to the problem of the nature of numbers. We see, first, that for an understanding of the meaning of numbers we must look to the 'game' which numbers play, that is to arithmetic. The numbers, one, two, three, and so on, are characters in the game of arithmetic, the pieces which play these characters are the numerals and what makes a sign the numeral of a particular number is the part which it plays, or as we may say in a form of words more suitable to the context, what constitute a sign the sign of a particular number are the transformation rules of the sign. It follows, therefore, that the object of oue study is NOT NUMBER ITSELF BUT THE TRANSFORMATION RULES OF THE NUMBER SIGNS.

Interseting, but debatable ...

More than 60 years before, Frege alredy criticized this view; see Gottlob Frege, Basic Laws of Arithmetic (1893), new english translation by Philip Ebert & Marcus Rossberg, Oxford UP 2013, page xiii :

[there is a] widespread tendency to accept only what can be sensed as being. [...] Now the objects of arithmetic, the numbers, are imperceptible; how to come to terms with this? Very simple! Declare the number-signs to be the numbers. [...] On occasion, it seems that the number-signs are regarded like chess pieces, and the so-called definitions like rules of the game. In that case the sign designates nothing, but is rather the thing itself. One small detail is overlooked in all this, of course; namely that a thought is expressed by means of "3^2 + 4^2 = 5^2", whereas a configuration of chess pieces says nothing.

Answer 8

In order to clarify Fine's definition of "number of thing", which is quite different form the "modern" set-theoretic approach, I think can be useful to refer it to the philosophical tradition of XIX century British empricism.

In particual, the philosopher John Stuart Mill devoted part of his work A System of Logic, Ratiocinative and Inductive (1843) to the discussion of the foundations of arithmetic.

Here some passages, which - I hope - can clarify Fine's definition :

Three pebbles in two separate parcels, and three pebbles in one parcel, do not make the same impression on our senses,- and the assertion that the very same pebbles may by an alteration of place and arrangement be made to produce either the one set of sensations or the other, though a very familiar proposition, is not an identical one. [...]

The fundamental truths of that science [the science of Numbers] all rest on the evidence of sense,- they are proved by showing to our eyes and our fingers that any number of objects, ten balls, for example, may by separation and rearrangement exhibit to our senses all the different sets of numbers the sum of which is equal to ten. (CW VII, 256-57)

Thus, when we say that the cube of 12 is 1782, what we affirm is this: that if, having a sufficient number of pebbles or of any other objects, we put them together into the particular sort of parcels or aggregates called twelves; and put together these themselves into similar collections,- and, finally, make up twelve of these largest parcels: the aggregate thus formed will be such a one as we call 1728; namely, that which (to take the most familiar of its modes of formation) may be made by joining the parcel called a thousand pebbles, the parcel called seven hundred pebbles, the parcel called twenty pebbles, and the parcel called eight pebbles. (CW VII: 611-12)

Mill's naturalistic approach to the foundations of arithmetic is based on the "basic" processes of joining and separating that give rise to and decompose "aggregates" of physical objects.

The empiricist view of Mill was sharply criticized by Gottlob Frege in his fundamental Die Grundlagen der Arithmetik (The Foundations of Arithmetic) (1884).

For an exposition of Mill's philosophy of mathematics see Philip Kitcher, Mill, mathematics, and the naturalist tradition, into John Skorupski (editor), The Cambridge Companion to Mill (1998), page 57-on.

Answer 9

In the book, the "number of things" is effectively distinct from their representation. Suppose you have guests you wish to invite to a party. What is the number of guests-things that you are inviting?

If you are inviting 5 friends, we'll call them John, Fred, Mary, Jill, and Barney. There are 5 guest-friend-things that you are inviting to the party.

But now, what if the party is a masquerade ball, and they're all in disguise. John is dressed as a ghost, Fred as a goblin, Mary as a witch, Jill as a pumpkin, and Barney as a dinosaur. Just because they are now ghost, goblin, witch, pumpkin, and dinosaur doesn't change the number of guest-friend-things that you have invited to the party. Their characteristics have changed - they no longer look like your friends, they look like their disguises.

What if the 5 of them come dressed all as indistinguishable ghosts. Does that mean that we say only one ghost has come to your party? No, because they can still be distinguished by their spacial locality, time of arrival, height, weight, sheet color, etc.

What if they wore the exact same costume and you never saw more than one at a time - such that there were no defining characteristics separating one friend from another. You might not be sure how many guest-friend-things you had at your party. THIS transformation has destroyed the distinctness that separated them prior to this, thus it is not a valid transform for enumerating the number of things.

The idea of "number of things" with respect to your invitations is specifically the property of the group such that any changes (relabling, renumbering, reordering, but NOT duplicating, eliminating, or counting subsets) that preserve distinctness of the elements maintains that property. It is not concerned with whether or not the value of that property is 1, 5, or a million-billion, only that the "number of things" is a finite value that keeps this property.

With regards to plain English, the number of things is just... the number of items of interest. It doesn't get any more simple than that, and because it's such a simple concept, it's very difficult to write a precise definition that does not cause issues in possible colloquial expressions.

Answer 10

This question (and many of the answers, for that matter) overlooks the purpose of mathematical theory, which is to treat axioms as something given. We assume that we have a notion of (for example) distinctness, and then explore the consequences of having this notion.

In other words, it is impossible to ask the question "How many elements are in the set $\{A,A,B\}$?" without first giving axioms about $A$ and $B$. According to standard mathematical syntax, we should really only ask this question after re-labeling to $\{A,A',B\}$ to avoid confusion, but this is a matter of communication and practicality, not dogma and certainly not some kind of truth about sets.

Mathematics, in the words of Roberto Unger, is a "visionary exploration of a simulacrum of the world". If you disagree with somebody else's vision, that's perfectly okay. But if you think you have an issue with mathematics itself, then chances are that you are generating your own contradictions by misusing language. If you are clear about what properties your notion of distinctness is supposed to have, then set theory applies, it is only a question of how. It is not prescribing a particular form of distinctness, but rather exploring the commonalities between all forms of distinctness.

Answer 11

It seems that the answer to your question is highly intertwined to what 'a thing' is. You might be aware that as abstract a question it might be, it has been asked repeatedly in the physics community in the context of quantum field theory and the foundations of quantum mechanics (see Paul Teller and Chris Isham, for instance). One of the conclusions is that the concept of a thing as an essence to which properties 'adhere' is to be rejected. This is what Teller describes as the problem with the 'labeled tensor product Hilbert space formalism', as it is incompatible with the physical behaviours that are actually observed. So if you want a universal definition of 'number of things' you can't avoid these considerations on what a thing is and on what distinguishability is from a physical point of view. (unless you want to a definition that applies to a universe that is not our own).

Just to give you an example, let's say you have one photon in your right hand and one in your left. You can distinguish them by referring to which hand they are in. So the 'number of ways of putting them in your pocket' is 2 (first the one in your left hand, then the one in your right hand or the other way around). However, once in the pocket, they become physically indistinguishable and 'the number of ways of taking them out' is 1 (out comes one, then the other).