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The first time I came across this distinction was when I asked this question. It was highlighted to me that there is a difference between the matrix and how we represent it. There is a difference between a number and how we represent it. In general, there is a difference between a mathematical object and its representation. I really am not able to wrap my head around this idea, even after giving it a lot of thought.

I am unable to think of numbers, matrices, vectors, and all related entities, by devoiding them of their notation. Even if I'm not writing down a matrix/vector/number on paper, when someone says 'a matrix' I'm basically drawing it in my mind. No concept of matrix far removed from its notation exists in my mind.

Could someone assist me to make sense of mathematical objects by freeing them from their notations, in detail?

I would be grateful for any resources as well that you could share.

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    The issue is that is difficult to "speak of" an object without using some sort of symbols... Thus, in principle, there is a clear difference between an object and its name. The same for numbers: IF we assume that they are some sort of (abstract) object, we have the number two and its names: "two", 2, 1+1, etc. Dec 20, 2022 at 13:34
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    But it is true that modern abstract mathematics is impossible without the appropriate symbols invented to describe them... Modern mathematics starts during the Renaissance with the invention of algebraic symbolism. Dec 20, 2022 at 13:36
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    Ordinary mathematical equations can be interpreted as computations. If we adopt the definitions of numerals (names for numbers): 1 = s(0) (the successor of zero) and 2=s(1), the equality 1+1=2 holds because when we evaluate the expression 1+1 according to the rules of arithmetic what we get is s(1). Thus, when in mathematics we state an expression like 1+1=2 (a string of mathematical symbols) we are using numerals to say something about numbers; specifically, we are stating that the result of the operation of adding one with one produces as result two. Dec 20, 2022 at 14:13
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    That you cannot think of numbers, matrices, vectors, without their notation should not create difficulties with differentiating them from it. When you are thinking of a matrix, then get distracted, then go back to thinking about it again, you have to redraw the same matrix in your mind. Therefore, the matrix is distinct from its notation, for the prior notation you drew for it is long gone. This is not unique to math. A shape too cannot be imagined without its material embodiment, but surely the two are distinct.
    – Conifold
    Dec 20, 2022 at 20:19
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    What's the difference between an apple and the world "apple?" One is a thing, and the other is a symbol for a thing. Now of course your question is, what is the nature of a "thing" that is entirely abstract? The symbols "4" and "2 + 2" both point to the same abstract thing, but what abstract thing is that? Is there really any thing that is being pointed to? Where does it live? Good questions.
    – user4894
    Dec 21, 2022 at 0:02

5 Answers 5

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If numbers are symbols, then which of these is the number six:

  1. six
  2. seis
  3. 6
  4. VI

If a number is identical to its notation, then you can't just say they all represent six because one bit of notation must actually be the number. Which is it? Which is the real number six?

I think you can see the problem here. The problem is that the notation is arbitrary and conventional, but the number is not. There is a single number that is the sum of 3 and 3 and is also the sum of 4 and 2. There is nothing arbitrary or conventional about that. It doesn't matter what notation you use to represent the number six, it only matters what the mathematical properties of six are.

I suspect that your difficulty is that you are trying to push other properties onto the number six because such properties are familiar to you. You are familiar with things having a location, for example, and a visible representation, so you want the number six to have those sorts of properties. But the number six has none of those properties. The only properties it has are mathematical ones. Don't try to layer other properties on top of it just because it is described as a thing.

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  • What are those mathematical properties of number six? Dec 23, 2022 at 16:36
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    @HarshitRajput, that it's equal to 3+3 and 4+2, that it's an even number, that it has two prime factors, etc. Dec 24, 2022 at 6:23
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    @HarshitRajput, this answer is a good one. Here's what I know, got it all from a book on elementary mathematics for teachers (that's the title, if you're interested). There's a difference between number and numeral and this fact maps perfectly onto yer question (a matrix as a mathematical object vs. the notation employed to represent & work on them). Numerals are the symbols "1", "9", etc., but Numbers are the concepts they, the numerals, denote. Dec 30, 2022 at 1:22
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    @AgentSmith I'm able to appreciate now the difference between a number and a numeral, I just started to read a book - Set theory and the structure of Arithmetic by Hamilton and Landin. I feel this is a good one, besides all the other excellent recommendations here. I would also check out the book you suggest. Dec 30, 2022 at 16:29
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    @HarshitRajput, I'll have to check. Dec 31, 2022 at 13:54
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Mathematical symbols are inherently abstract, as they are a way for one to represent the concrete through a "neutral" way. As such, the concept of the number "six" should not be familiar to you, nor would you be able to represent it in your mind. You may consider 6 points, 6 apples, even 6 boxes, but the abstract concept of a number is unreachable.

It is precisely because of this that we instead depend on notations, as the concepts become too difficult for us to even use analogies. For example, mutliplication can be seen as taking a certain number of packs of apples. However, complex exponentiation will be difficult to imagine or picture without notation.

As such, we may consider mathematical objects to be intricately tied to their representation, as this representation is the only way we may picture them while preserving all the information we have about them. Unlike languages where the words themselves are a way to express an ideal related to an object, mathematical symbols are entirely abstract, and so we need a way to picture them.

Like said previously by David Gudeman, the representation itself is arbitrary. However, such a notation is also necessary, because we simply cannot perceive it in a way that would make sense in every case. For example, to imagine 5 bananas is okay, but 5^3 is difficult to imagine. "Bananas cubed" is not a unit which we can understand, and so we use the notation. Of course, we could have written it in binary, 101^11, but the actual content remains the same.

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Some random thoughts, let's discuss the concept of a number specifically:

  1. The concept was in use long before math notation was created, there were whole civilizations that had no form of writing and math, where people owned property, traded goods and services with each other, paid taxes etc.

  2. As some people pointed out, there are many notations for representing numbers starting from the ones that don't have the concept of a zero, modern number system with different bases etc.

  3. By the way, the concept of a natural number is a byproduct of counting, so number was and still is mainly "something we can count with", not a mathematical object - this is why we use base 10.

  4. There are also many types of numbers, some of which use the same notation e.g. when you see 1, are you thinking of the natural number 1, the integer 1, the rational number 1 etc. of course there are ways to represent all those, but my point is that notation is often ambiguous.

  5. If the number is not the symbol, what is it? The question is actually far from obvious (it's one of the most fundamental philosophical and mathematical questions to ever been asked). So a number is much more than the notation. Reading: https://www.cantorsparadise.com/freges-concept-of-natural-numbers-1e8e70a23a67

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It sounds like you are confused about the distinction between use and mention. There is a distinction between a number (as a concept) and the notation used to represent that number. As pointed out in the other answer here, a single concept might even be represented by several alternative forms of notation.

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You might not have to think of anything of abstract objects devoid of their properties if you give weight to some sort of bundle theory of objects(REP). While there is a lot of support for neo-Platonic thinking, there are empirical and constructivist accounts that see math objects as nothing more than references to abstractions of physical objects or even other abstractions.

This is a problem in the philosophy of mathematics. According to Michael Dummett in his Frege: Philosophy of Mathematics the founder of analytic philosophy, Frege was engaged in exchanges with Dedekind, Husserl, and others about this very question in regards to what a number is. In German, he was able to capitalize on a divergence in lexemes Anzahl und Nummer where the former might be translated as quantity as in conceptual cardinality as opposed to number which is often used as a synonym for digits in English. Therefore, if mathematicians realize a dichotomy between the general and the specific, what some philosophers refer to as universal and particular, then cardinality is reified by collections of digits, whatever there representation (these are known as graphemes in the philosophy of language). Today, this distinction is generally subsumed by philosophers of language between the dichotomy between syntax and semantics, how something is represented versus what it means. This is kith and kin to, for instance, the difference between a sentence and a proposition. 'I am hungry' and 'Hunger has beset me' are synonyms.

What you ask about is a bit deeper philosophically, because it is an ontological question about what it means to exist, and there's a whole conversation in the canon (relatively) recently among Meinong, Carnap, Quine, and other famous folks that tries to hash it out. Stanford has an introductory article on abstract objects.

My take on it as a constructivist is that references can refer to physical objects, mental objects, or other references. Mental objects are mental representations of various inutitional (psychological) faculties abstracted from instances.

Thus, we abstract from a collection of fishes, say trout, perch, salmon, and bluegill, that are subitized or counted, because we can differentiate inuitively between species as prima facie natural kinds and we can metaphorically contain those species as sets, classes, what have you and reference with a category label. Right? In our childhood, we could have gathered blocks shaped as triangles, squares, circles, and hexagons, counted each, done operations on like sets of collections of shapes. The abstract matrix object, therefore, is just a reference to the process and the syntax we use during this activity to communicate with others. Thus, there is a utility and the linguistic behavior is preserved and used in what Wittgenstein would call a language-game. In a nod to our primary sysadmin, such abilities are rooted in faculties of collective intentionality as described by M. Tomasello.

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