0, 1, and beyond
Timeline
I decided to start writing a book called "Introduction to Mathematics" on 21st of August 2019. This book was supposed to introduce fundamental mathematical concepts to a layman who never really explicitly studied any math. However, of course, it had to explain not any mathematical concepts but my mathematical concepts. Here is the table of contents of the unfinished book:
- Object
- Number
- Operations
- Expression and Function
- Set
- Relativity
- Ball Experiment
- Logic and Memory
- Continuity and Movement
- Classification
I stopped writing it for a while before making some developments again on February and August 2020. To give a brief overview of the "finished" (but also clearly draft) parts of the book, I talked about data (or information) that constitutes every observable object; then I tried to use the notion of this "objectness" to define numbers; addition, subtraction, and multiplication/division were of interest in the following chapter of operations; I defined and distinguished the concepts of an expression and a function by introducing the substitution operation; then a set was defined as a collection of any type of data; by jumping to the classification, I talked about assembling and disassembling objects by analyzing information (or the so-called data particles) that constitute them. Unfortunately, when I look back at the chapters that I didn't even start writing I don't exactly remember what was my thought process and what I was exactly thinking writing about at that moment when I named them. Good part is that at least I have some (biased) guesses.
In this blog post, I wish to bring back some of the concepts such as objects, numbers and operations from that book and add a little bit of retrospective window to it. Please note that in the following section and its subsections, I may not only modify my old notations a bit when convenient but I may also use some set-theoretic notions and notations to illustrate the similarities between the two -- my concepts and set-theoretic concepts.
Introduction to Mathematics
Object
When we think of an object in the real world we certainly do think of its intrinsic characteristics. However, this is not the only thing that we think of -- we also (even maybe unintentionally and forcefully) think of its surroundings. For example, when we look at an apple sitting on a table, we may focus on the apple (and its properties) as our main object and table as its surroundings (or its supportive evidence). This natural intuition has brought a new notion of an object which is given as follows: an object is something defined by its primary and secondary definitions.
$$ \forall O\ \text{Object}(O) \iff \exists P, S\ \text{Definition}(p_{\in P}) \land \text{Definition}(s_{\in S}) \land O = (P, S) $$
Here $p$ describes the intrinsic characteristics of the object and $s$ describes its extrinsic or "place-like" characteristics. To bring back the "an apple on the table" example, although we could zoom in on the apple to only visually see it and nothing else, then there are two possibilities:
- We could pick some part of the apple as $p$ and the rest as $s$, and therefore, lose the notion of the apple as a whole.
- We could pick the whole apple as $p$ and leave $s$ empty since there is nothing left to be picked, and therefore, lose the notion of its identity.
Number
First, we need to acknowledge the following axiom:
$$ \forall O = (P, S)\ \text{Object}(O) \iff \exists \top\ (\top \subset P) $$
which states that there exists a presence of any kind of observable property (i.e., $\top = \{\varnothing\}$ and therefore, $P = \{p, \varnothing\}$ but $\varnothing$ will be omitted to reduce repetition) that is common in all places. Let's look at some numbers then:
$$ 0 = (\{p\}, \varnothing) $$
$$ 1 = (\{p\}, \{s\}) $$
$$ 2 = (\{p\}, \{s, s^\prime\}) $$
For example, the number 2's object definition states that there exists a $p$-characteristic object in $s$-characteristic place and another $p$-characteristic object in $s^\prime$-characteristic place. $p$ and $s$ could be any sentence in the first-order logic such as $p = \text{Red}(x) \land \text{Round}(x)$ and $s = \text{OnTable}(x)$ for the number $n(x)$ (note that $x$ here refers to an "apple").
Pure numbers
Besides representing grounded or impure numbers, we can also represent the basic fundamental existence (i.e., the presence of anything) by using the commonality-characteristic $\top$.
$$ 0 = (\top, \varnothing) $$
$$ 1 = (\top, \{s\}) $$
$$ 2 = (\top, \{s, s^\prime\}) $$
For example, the (pure) number 2's object definition states that there exists something in $s$-characteristic place and something in $s^\prime$-characteristic place. Although this looks a bit similar to the impure numbers, the difference is how we have abstracted away the primary definitions of objects, and now every pure number is similar to its secondary definition, i.e., $n \sim s(n)$ where $s(n)$ is the secondary definition of number $n$ (and for this matter, $p(n)$ is the primary definition of $n$).
Operations
The essence of any operation is a consequence of acting or operating on a piece of information. In this section, I will use $O = (P, S)$ notation to indicate that $P$ and $S$ are actually sets of primary and secondary definitions of the object $O$ respectively.
Defining addition
Adding some number of something to some other number of the same thing is intuitive and straightforward thing to understand. Addition on two such numbers is defined as shown below:
$$ (P, S) + (P, S^\prime) = (P, S \cup S^\prime) $$
where the primary definition obviously remains the same and the secondary definition of the new number is now logically interpreted as $\bigwedge_{s \in S \cap S^\prime} \texttt{id}_{P}(s)$ where $\texttt{id}_{P}(s) \equiv \text{'}P\ \text{is identified by}\ s\text{'}$. Here single quotation marks ($\text{'...'}$) mean that we do not have to evaluate the expression in between logically, and instead, it must be evaluated syntactically (i.e., by symbolic substitution).
There is no equivalent of $(P, S) + (P^\prime, S)$ and $(P, S) + (P^\prime, S^\prime)$ since the primary definitions are different for the two numbers, so adding them does not make sense. However, one could make the following case:
$$ (P, S) + (P^\prime, S^\prime) = (P \cup P^\prime, S \cup S^\prime) $$
where the primary definition of the new number is logically interpreted as $\bigvee_{p \in P \cup P^\prime} p$ (where $\varnothing$ is interpreted as false). Even though this could be possible, the following addition still would not make sense unless we combine the characteristics altogether:
$$ (P, S) + (P^\prime, S) = (\bigvee_{p \in P} p \land \bigvee_{p^\prime \in P^\prime} p^\prime, S) = (\overline{\overline{P} \cap \overline{P^\prime}}, S) $$
where $\overline{P}$ and $\overline{P^\prime}$ are the complementary sets of $P$ and $P^\prime$ respectively (if such a thing exists).
Defining subtraction
Similar to the addition, it is relatively easy to define subtraction on numbers with the same properties. Subtraction on such numbers is defined as follows:
$$ (P, S) - (P, S^\prime) = (P, S \setminus S^\prime) $$
whose secondary definition is interpreted as
$$ \bigwedge_{s \in S} (s \in S^\prime \iff \lnot \texttt{id}_{P}(s)) \models \forall s \in S\ \forall s^\prime \in S^\prime\ s \neq s^\prime \implies \texttt{id}_{P}(s) $$
The entailed sentence perfectly matches the true sentence directly derived from the definition of the number which is $\bigwedge_{s \in S \setminus S^\prime} s \iff \forall s \in S\ \forall s^\prime \in S^\prime\ s \neq s^\prime \implies \texttt{id}_{P}(s)$ where $=$ is treated as an element-wise syntactic equality.
However, what would it mean to subtract one apple from two cars? It would mean exactly nothing and so, subtraction on such numbers are defined as shown below:
$$ (P, S) - (P^\prime, S^\prime) = (\varnothing, S \setminus S^\prime) $$
The primary definition being the empty set (i.e., $\varnothing$) does, in fact, tell us that there is no such grounded or impure number defined on such a subtraction.
Defining multiplication
When multiplying two numbers, one needs the semantics concerning the operands. That is to say, one of the operands needs to encapsulate the objectness and the other one amount of repetition (of addition or existence of the objectness). For this reason, let the left operant to denote a grounded number and the right operand to denote a pure number. Now, let's define the multiplication as shown below:
$$ (P, S) \cdot (\top, S^\prime) = (P, S \times S^\prime) $$
where $P \times \top \sim P$. Note that it should be obvious that to obtain a pure number as the result, one could easily use two pure numbers as the operands, and in that way the multiplication operation would also make sense for pure numbers too.
While it would make sense to multiply any type of number (i.e., grounded or pure) by a pure number, it does not simply make sense to multiply two grounded numbers. One obvious reason for this would be the following: what would you expect 3 cars multiplied by 2 apples to mean and be equal? That's why it just does not make any sense.
Retrospection
Pure numbers, according to my definition, omit the notion of primary definition, however, they keep the notion of distinct secondary definitions. So, two pure numbers with $n$ distinct primary definitions are still distinct numbers even though they represent the same quantity $n$ (e.g., 3 fireplaces versus 3 garages). If we omitted the secondary definitions too, we could call them "the purest numbers". We could do so by writing
$$ 0 = (\top, \varnothing) $$
$$ 1 = (\top, \top) $$
$$ 2 = (\top, \{\varnothing, \top\}) $$
Such numbers are indeed very similar to the numbers defined by John von Neumann in the elementary set theory:
$$ n^+ = n \cup \{n\} \sim \{s_1, \cdots, s_{n+1}\} $$
But what if I just wanted to add two numbers up? Since the secondary definition of bigger or equal number would include the secondary definition of the other, the result would be equal to that bigger or equal number all the time (e.g., $1 + 1 = 1$). To prevent this from happening, we could always assume the following:
$$ \forall n, m\ s(n + m) = s(n) \times \{0\} \cup s(m) \times \{1\} $$
$$\ \forall n, m\ s(n - m) = s(\hat{n}) \setminus s(\hat{m}) $$
where $s(\hat{m}) \subset s(\hat{n})$ and there exists a pair of bijections such that $\hat{n} \sim n$ and $\hat{m} \sim m$. Although it is trivial to do the same for the primary definitions in this case, I would like to write them down as well:
$$ \forall n, m\ p(n + m) = \{x \vert x \in p(n) \lor x \in p(m)\} $$
$$ \forall n, m\ p(n - m) = \{x \vert (p(n) = p(m) \iff x \in p(n)) \land (p(n) \neq p(m) \iff x \in \varnothing)\} $$
second of which means that $p(n-m) = p(n)$ if and only if p(n) = p(m) and $p(n-m) = \varnothing$ otherwise:
$$ \forall x\ x \in p(n-m) \iff [(p(n) = p(m) \iff x \in p(n)) \land (p(n) \neq p(m) \iff x \in \varnothing)] $$
Last but not least, if one wanted to capture the existence of an object, say, in the first-order logic then it could be done by constructing a sentence which contains the conjunction of all the descriptions (or first-order sentences for this matter) used in the primary and secondary definitions of an object predicated on the object itself.
$$ \bigwedge_{p \in P} p(\text{'}(P, S)\text{'}) \land \bigwedge_{s \in S} s(\text{'}(P, S)\text{'}) $$
Here is an example for the "an apple on the table" object:
$$ \text{Red}(\textit{RedAndRoundOnTable}) \land \text{Round}(\textit{RedAndRoundOnTable} \land \text{OnTable}(\textit{RedAndRoundOnTable}) $$
Conclusion
I spent significant amount of time while I was developing these concepts from scratch and I definitely enjoyed it very much. During the process, I also realized that it is really not that easy to understand numbers and operations (I used to tease one of my group mates by asking him whether he believes that he could really understand the wisdom of addition), and what they are operating on in our imaginary world (i.e., objects).
I still find quite interesting ideas when I look back at this probably underdeveloped concepts of mine. While the general notions of objectness, numbers, and operations are hopefully insightful and intuitive, there is also beauty in the technical details. For an example, it sounds somewhat philosophical for the commonality property $\top$ to exist in all perceivable objects. However, almost jokingly, its only element $\varnothing$ is what defines a non-object (by invalidity or absurdity).
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