Mathematical objects existing as different instances

Question

I have a slightly complex conceptual question about the idea of 'multiple' instances of mathematical objects. In particular Real Numbers, and generally the idea of having multiple instances of conceptual objects my main idea in this is that recently I based my understanding of mathematical objects as being sort of conceptual objects that you can reference in expressions and whose properties we can talk about. The idea of sets ties in with this quite nicely (there's only one object, there's only one idea, writing '2' is just a reference to two and therefore the set {1,2,3} contains the same three conceptual objects and any set with these objects in them has to be the same object again, but the idea of multisets has changed this for me and I do not quite understand this idea of 'multiple entities' of the same object. I've also noticed is with digits, we define a specific digit as being a specific symbol, yet, we will call a numeral with the same digit twice as a 'two-digit number' this is strange to me based on the definition of the digits as a class of different but perhaps this is a shared property of all symbols, letters, etc., but also perhaps this has some relation with the concepts I am trying to deal with, keep in mind, I am very early in my studies and perhaps don't have the most solid rigorous understanding of higher mathematical theories, in particular the idea of variables and quantities.

Do we consider different quantities to be different instances of the same number?

Answer 1

Short Answer

The answer to your question somewhat depends on your philosophy of mathematics. Your observation that there seems to be particulars and universals in numbers and other math ideas goes back to Plato whose thinking is still widely influential among mathematicians. Ultimately your question is ontological, because it asks about the nature of the being of numbers. We can use mathematical formalism as an example of how someone might provide a specific answer to the question of 'Are the 2's in 2+2 the same?'

Long Answer

The question you ask is a fundamental question in the philosophy of mathematics. 'What are numbers?', 'What are shapes?', etc. don't have simple answers like addition problems. For a Platonic thinker, individual numbers we use in discourse are imperfect copies of Forms. For a formalist, a number is simply a symbol that is defined by other symbols and neither really has much in the way of meaning. A mathematical structuralist defines a number in systems of relations, for instance, appealing to Peano's Axioms to define the natural numbers. In computer science, which lends itself to mathematical constructism, mathematical objects are objects instantiated from classes by constructors. All of these are examples of how one number might be considered an instance of something more general.

What you should take away from the response is that there is no broadly accepted consensus to your question with philosophers of mathematics taking a wildly diverse array of positions. According to J.R. Lucas, pure mathematicians are said to gravitate towards platonic thinking (SEP) which holds that there is another physical-like realm our minds are in touch with. Note, that once you begin talking about our universe, our minds, and other universes of discourse, you are firmly in philosophical territory. On the other end of the spectrum, mathematics can be taken to be intuitional as argued by Brouwer or empirical as argued by Mill.

On the other end of the spectrum, recent arguments from embodied cognition such as those made by Lakoff and Nuñez in Where Mathematics Comes From reject Platonic thinking entirely, and essentially argue that Platonic thinking is nothing more than arguments built on the fallacy of reification.

My personal biases are to reject Platonic thinking as unscientific in the empirical sense. Until someone sends a probe to the Realm of Forms and shows me some false-color images of these forms, I presume they have connected to the brain much in the same way data types are related to microprocessors. Whether or not that fits your Weltanschaung is a matter for you to decide. But whatever your decision, be skeptical of anyone who is sure they have the one, true answer. They're more likely to be true-believer than a critical thinker. In modern epistemology, fallibilism is the broadly held consensus.

Formalist Example of How 2's Are the Same and Different Simultaneously

Are the 2's in 2+2 the same? Maybe. Let's see what a formalist would say.

I see you are interested, from the comments on user4894's response in multisets. Here we can give an example of how a formalist might provide a response to your question 'Are we adding the same number in 2 + 2, or are they somehow different?'.

First, note the existence of the sequence of naturals by intuition (0,1,2,...). At this point, we have an intuitive sense that 2 is defined as the cardinality of a set such as {a,b}, but a formalist goes a step further and looks to justify the existence of all naturals N and explain how 2's can be the same and different at the same time. This can be accomplished with Peano's Axioms (PA).

Briefly, there are a few rules we can use to define N. We can assume 0 exists. We can assume a successor function exists, written as (x), so that 1:=(0), that is, 1 follows 0. Then, 2 is simply the successor of 1, or (1), which by definition is also ((0)), and so on. Now, addition is simply defined as the sum being nothing but the addend defined as the successor of the adder. Hence, 2 + 2, is ((x)):x=((0)) where 2+2 is equivalent to ((((0)))) which is a good thing, because by definition, the fourth successor of 0 is 4! Now note, technically speaking, 2+2 can be abstracted as a sequence itself: (2,2). But even though both the first 2 and the second 2 are ((0)) by PA, they are not the same 2 when considering order. In fact, implicit in (2,2) is the idea that (2,2) is really (2,2*). How so? Because any sequence by definition is a set mapped onto the naturals, that is, we can pair (2,2*)->(0,1).

So, are they the same? Yes, both 2's in 2+2 are defined as ((0)). Are they different? Yes, both 2's occur in different locations of the sequence (2,2*). So they are the same and different. This seems like a contradiction, but we simply have to provide context, an idea that is important in a logical concept known as dialetheism.

Now, one might ask, are there rules for what we just did, and for a long time, mathematicians struggled to find them. But following Frege's original goals in formalism, ZF was introduced. This is called 'grounding set theory' in logic and is an example of mathematical foundationalism. In fact, we might not want to use ZF but prefer ZFC instead. Or maybe NBG is even a better foundation. Some eschew sets and look towards category theory. Important to our discussion is that we are following some formal rules.

Remember that successor function we presumed? It's an instance of the axiom of infinity. Are we allowed to differentiate the two 2's? Sure, as long as we accept things have order and the axiom of extensionality holds. This idea in mathematical logic is known as grounding, where we show that one set of mathematical truths used to prove things, like arithmetic, is contained or grounded in the another set of mathematical truths that prove that our first set doesn't create any contradictions.

Answer 2

This kind of thing comes up in formal math all the time. For example say we have the set of integers Z, and we wish to form the Cartesian product Z × Z, the set of ordered pairs (n,m) where n and m are integers. Nobody ever thinks twice about doing this.

But suppose we do think twice. Suppose, based on your question, we ask ourselves where exactly do we get two copies of the set Z? After all, a set is entirely characterized by its elements. There is exactly one and only one set of integers Z. How do we blithely invoke a second instance of it?

There's an answer. If someone ever challenges us, we note that given a set like Z, we can form the Cartesian product of Z with the singleton set {0}; that is, Z × {0}. This is the set consisting of all the ordered pairs (n, 0), where n is an integer. This set is distinct from the integers, even though it's in obvious bijective correspondence with the integers and may, if we cared to do so, be taken as isomorphic with the integers and their usual operations of addition and multiplication, and their usual order relation.

Now we take the same set Z and form the Cartesian product Z × {1}, the set of pairs (n, 1) for every integer n.

We now have two distinct sets, Z × {0} and Z × {1}, each of which exists within set theory, which are distinct from each other, and which can each be taken as a proxy for the set of integers. Conceptually we take one copy of the integers and "paint it blue," and another copy and "paint it red." So now we have two distinct sets that can form the Cartesian product (Z × {0}) × (Z × {1}), which for obvious convenience we consider to be Z × Z.

In this way, given any mathematical object of which there is only one, we can always whip up a copy of it which is distinct from the original, yet behaves in all important respects just like the original. The reason this works is that within set theory, every mathematical object is ultimately a set, and we are always allowed to form Cartesian products with other sets like {0} and {1}.

You will rarely if ever see this procedure written down in a math book. But this is in fact exactly what a mathematician, or at least a set theorist, would reply if asked how to justify an operation that involved "two copies" of the same set.

A familiar example is the standard Cartesian plane made up of "two copies" of the real numbers displayed at right angles to each other. Every high school student is familiar with this idea. Nobody ever asks where we got a second copy of the set of real numbers, since there is actually only one set of real numbers within set theory. There is no principle or rule of set theory that allows you to duplicate identical copies of sets.

The answer is that secretly, we use this trick to make as many identical copies of the real numbers as we need.

If we're challenged, as in this question, we can explain how it's done formally. In practice nobody ever asks and we never need to air our dirty set-theoretic linen.

Answer 3

You can visualize a multiset as a function mapping elements to natural numbers. That is, the multiset {0,1,1,1} would be represented by the function that maps 0->1, 1->3, and every other possible element maps to 0.

Or, you can take into account where multisets come from, and visualize them as a mapping of slots to elements. For example, one common example of a multiset in mathematics is the multiset of the prime factors of a number. The factors of 24, for example make up the multiset {2,2,2,3}. This doesn't mean that to factor 24, you need three copies of 2, it only means that you can divide 24 by 2, and then divide 12 by 2, and then divide 6 by 2.

An alternative way to list the prime factors would be to actually list them: first divide by the smallest prime number that evenly divides the number, then divide by the smallest prime number that evenly divides the result, etc. Doing this, you generate the list [2,2,2,3]. This doesn't require multiple copies of 2 either; it's a structure with 4 positions or slots, and each slot is associated with a number. There is no reason why two slots can't be associated with the same number.

So remove the ordering from the list [2,2,2,3] because the ordering is not relevant to the problem, and you now have the multiset {2,2,2,3}. But because of where it came from, we can see that we basically have four slots, and each of those slots is associated with a number. It's just that now, we are not putting the slots in order.