Reflections on Mathematics II — Defining functions for the Ideal Space
Attention: Explicit content. This article may run psychological warfare on you. But in the end, I am sure it will be good for your psyche.
TL;DR
The article provides some proofs and theorems without providing any formal laguage. The formal mathematical language is quite young compared to math itself and is not able to formulate every problem properly. The argumentation finally concludes multiplication cant be a mathematical function.
Introduction
This is the more technical and interesting part of my reflection for the first semester of mathematics. I dissect basic structures with only the knowledge I can understand with 1st semester knowledge and explore if it is possible to understand the world better.
I was told from another student that I first have to repeat what others say in order to be allowed to do my own math. And exactly this is part of the street lore that is slowly corrodes our society. Mathematics is free and independent. It does not depend on norms and social rules or politics.
In the first article, we argued that good mathematics is about finding functions from an ideal space into another space, for example a natural language or a mathematical structure.
What is a function?
In mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y.
What are algebraic strcutures?
Algebraic structures are n-Tupel that consist of a set and some operations. There are all kind of structures but the most common are sets, group-like structures and fields. For more insight on the topic read the Wikipedia article.
There is a hierarchy associated with each structure. For example a set is lowest in hierarchy as all elements are degenerate. A higher order structure for example is a monoid. Each of these structures have their own properties and axioms attached to them.
Constructing functions for structures
Warning:
These thoughts seem wild, unrelated, and unclear, but we have to understand that my thought process is not entirely conscious. That is parts of my subconscious speak there, too. We later use the rules of mathematics to make sense of these.
Currently, my hypothesis is that this map (function) is surjective. A surjective function is a function that can map all elements of Y from X. This should be clear because:
A: It must be a function, as objects in the ideal space that can’t have a manifestation would not make sense. So, all objects in the ideal space must have at least one manifestation (see above reasoning from Gödel). This could be the object itself (it is manifested as it is). We don’t know. Not relevant. But there can be multiple manifestations.
Q: If there are multiple manifestations of an object from the ideal space are can they be in the same picture set? Or can an ideal object manifest two different objects in the same set?
A: Intuitively, yes (corresponds to no function so not surjective), but unintuively no. For example, if I have to change axioms to do a different manifestation, then the object would be in a different set.
Q: Can I change axioms and still be in the same set?
A: Yes, if the picture set is larger than the set of manifestations with that particular axiom set, but it would not make sense. So, every element of the picture set is a unique manifestation (it is not possible to manifest two different objects from one ideal object in the same axiomatic). If you add or subtract axioms to the set of possible axioms for a given picture set, you change the set.
Q: But are the two manifestations then really a contradiction?
A: If there are two objects that are different manifestations and in the same set, then there does not have to be a contradiction either. They could use different subsets of axioms and still be enough (compare Rings, monoids, etc). This hints that the thought structure is not clear yet, but one really has to distinguish the underlying and picture sets properly. A Group is a monoid etc. So, there can’t be two manifestations with the same axioms. Such as there is no contradiction.
In his proof, Gödel uses omega-consistency, which is on the top view a concept from logic. However, and this is astonishing to people new to higher mathematics, is that logic has flaws. This is what Gödel was showing in the first case. The surprising fact is that flaws in logic do not prove that mathematics has flaws. From my perspective, all boils down to finding good functions (maps).
Deductions from the reflection
All objects in the realm of manifestations are manifestations of an object in the ideal space. However, this was the definition of a manifestation or the whole space of manifestations.
If there is no contradiction, but the function is not injective, then the axiomatic is incomplete. I assumed, the function was surjective then if the cardinality of the picture set is smaller than the cardinality of the idea set, different ideal structures can manifest the same object in the picture set. And if it is a function, all ideas manifest. Yet, there is no contradiction.
- If the picture set and the ideal set are of the same size and the manifestation is a function, then there can be a function such that all ideas are mapped to the manifestation space in a way that there is no contradiction and they are complete. (Proof that when two sets are of the same size, the relation of injective, surjective and bijective is special in that sense, i.e., the function must be bijective).
- Either the manifestation of the ideal space is a function, then the picture space might not be complete but can be complete. It can’t contradict itself, though, because it is a function.
- Or if the manifestation is not a function then, there can be a contradiction. But if it was complete, then there has to be a contradiction (it may break here) or not all ideas are mapped. But if not all ideas are mapped it would not make sense as they would not be ideas. This would mean they would manifest different objects with different axioms. This in turn means, for example, that a Monoid would manifest a group and (both) a semigroup with an identity element. However, would that make sense? I can prove that would be wrong in the given axiomatic. Try to think these through again.
In my opinion, the function is surjective. But it may not be a function at all. On the other hand, if it wasn’t I would not produce logical results and the case would not make too much sense. Or formulated differently: The mathematician did not work hard enough.
Thus there must be a function that maps sets of algebraic structures to each other. Mapping from the full set of structures to another subset that function must be surjective.
A practical example for such functions
Interesting related proof from a practical session at the institute:
Q: What happens if you relate two different algebraic structures, e.g. vector and scalar, matrix and vector, natural number and the identity element of a field?
A: You said that, for example, the identity of the natural number, eg. 1, and that of the field are not the same, my subconscious says it must be the same somehow.
I think that the notation it that case is flawed. This is because you can only relate elements of a field if they are parts of the field, and otherwise the relation is not defined in the field and would not make sense.
A field is a triple of a set and two operations with certain axioms. Always. It can’t have three operations else it would create another structure, i.e., ordered field, which is a 4-Tuple. A relation and a set form an algebraic structure. Each field has only one identity element 1 regarding the second relation.
Thus, the identity of the natural number is becoming the identity of the other structure. For example, if you relate the 1 from the natural numbers with matrices, it is equivalent to writing an operation with the neutral element of the field of interest. The proof is easy, and maybe I will provide it in another article. This article shall be free of confusing notations, such that it remains purer.
E.g., with NxN-matrices, multiplying with 1 from the natural numbers is equivalent to multiplying with the unity matrix.
Q: Can 0 be equal to 1 in a monoid and higher structures? Respective what are natural numbers and does a 0 have any importance of sets with cardinality greater than 0 and greater than 1?
A: Of course not. Addition and multiplication are two different operations (functions). Intuitively, the multiplication is less powerful. The addition does not cause so many problems.
The introduction of inverses in higher structures makes everything so hard. Especially, division by zero is what makes everything that follows so complicated and makes analysis fail often. Intuitively the operations can’t be the same, as the neutral element does not relate to itself. Thus, the monoids (N, +,0) and (N, *, 1) are different as the cardinality of the first is one larger than the former.
Q: Why is it necessary to distinguish both?
A: Because it is clear that the operation of multiplication with 0 does not really help the structure much. For example if the set we are investigating is of cardinality one, it would look either like {1} or {0}. Assume that multiplication is defined on sets of cardinality 1. Then, starting from addition the smallest natural number is 0. Multiplying 0 with itself is yielding 0. Then, doing the same with 1 also yields 1.
Doing this with addition shows that if we add 1 to 1 the result would not be inside the original set as this would yield 2. Adding 0 to 0 yields 0 again and the result would be defined within the set. Applying the principle of induction we can now construct the natural numbers by adding 1 to every new element we yield by addition.
However, trying to do this via multiplication is not possible, because if we are strict, the multiplication is not a function.
Thus, the 1 can only matter if there are at least two numbers in the set. This is actually really important when defining functions on the natural numbers (e.g sequences and series — see part III for more in depth analysis of the phenomenon and real deduction of new numbers that are a subset of the natural numbers with the same properties but the additional property to approximate a circle).
Depending on the operation the number of a set with cardinality 1 is either 0 (addition) or 1 (multiplication). Yet, the multiplication is the summary of the addition done many times.
So in fact, it must always be true that the cardinality of a set forming a structure with multiplication is smaller than the set forming a structure with addition. Or how can a structure formed by mulitplication be larger than a structure formed by addition when considering the same basis set. It is not possible as the multiplication is the summary of the addition.
Q: Is multiplication a function?
In German, we say the operation and you have to enjoy this is eindeutig. The standard translation says the operation is not well-defined. This is a bad translation. The word eindeutig literally means the operation is not definitely pointing to one. You will see in part III why this is the case. Thus, I think German should become the language of science again.
Multiplication on the natural numbers is not eindeutig.
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