Is it possible to express all mathematics through atomic pieces of computation?

[tanus5]

I have been considering the idea for awhile that all mathematics could be broken down into atomic pieces that are indivisible, and therefore any formula could be determined through a process of elimination by trying every possible mathematical formula.

Here is what I have come up with so far, meant to be processed by a computer, but since I don't know everything about mathematics I don't know if every possible mathematical formula on the set of real numbers can be expressed with this method.

Every atom would be made up of the following 3 units

1. scaler {-∞,...,∞)
2. type { constant, variable, group/absolute value, floor/ceil value}
3. operation { + , * , ^ (power) }

In the case of a constant the scaler is that constant
In the case of a variable the scaler is multiplied by the variable
In the case of a group the absolute value of the scaler is the number of items in the group and the sign determines if it is a simple group or an absolute value
In the case of a floor the scaler is the number of items in the group and the sign determines if it is a floor or ceil

The reasoning behind the operations is that subtraction is expressed by a negative scaler, division can be represented by x^-1 * Y, and square roots can also be represented by powers.

The method of executing these operations would be by calculating the inner-most groups first and disregarding the operation from the first atom in each group such that "x + 1" could be represented by the 3x2 matrix

1 variable *
1 constant +

The operation on the first row of the matrix would therefore always be ignored, and in theory, a 3xN matrix could then define every possible mathematical formula.

I haven't included any calculus methods since sums can be expanded out and I believe integrals and derivatives can still be expressed with these atomic functions, but I could be wrong which is why I'm asking.

Using only standard variables (ie. no vectors, matrix, tensors, etc.) are there any other operations or features which should be included that are computable with finite computational resources?

[edit] Please note, a computer would solve this problem as a genetic algorithm since it wouldn't be possible to try every possible combination of scalars since there are infinite values that can be selected. Fitness values could be used to extract probable solutions and optimizations could be run to find the optimal constant values.

 

[pbuk]

Google theory of computation - the Wikipedia article is an OK start.

 

[tanus5]

@MrAnchovy,

Thanks I think that was exactly what I was looking for. It looks like I need to do more research into the turing machine also. What I was able to gather from the wikipedia article though is that there are problems that are unsolvable so before going down a rabbit hole to never return I'll also need to learn more about determining if a problem is solvable. There is no point in devoting computational resources to problems that can't be solved.

 

[pbuk]

You should also look at Mathematica/Wolfram Alpha, Maple, MATLAB etc. which have had many lifetimes of effort put into them.

 

[blue_raver22]

I find this idea intriguing and thought-provoking. However, there are a few points that need to be addressed in order to fully evaluate the concept of "genes of computation."

Firstly, the concept of breaking down all mathematics into atomic pieces is not a new one. In fact, this idea has been explored in various fields such as computer science, mathematics, and philosophy. One example is the concept of "primitive recursive functions" in mathematics, which are basic building blocks from which all computable functions can be derived. However, there is still debate about whether these building blocks are truly the most fundamental and whether they can fully encapsulate all mathematical concepts.

Secondly, the idea of using a process of elimination to determine every possible mathematical formula relies heavily on the assumption that there is a finite set of formulas that can be expressed. This is a bold claim and would require extensive proof and evidence to support it. Additionally, as you mentioned, the use of genetic algorithms to solve this problem would also require careful consideration and testing to ensure its effectiveness.

Furthermore, the proposed atomic units of computation (scaler, type, operation) may not be sufficient to fully encompass all mathematical concepts. For example, what about transcendental numbers or irrational numbers? These cannot be represented as a finite decimal or fraction and may require additional units to be included in the atomic structure.

In terms of your question about including other operations or features, I believe that the inclusion of calculus methods would be necessary in order to fully capture all mathematical concepts. As you mentioned, integrals and derivatives can still be expressed using powers, but there may be more efficient ways to represent them.

In conclusion, while the concept of "genes of computation" is thought-provoking, there are still many unanswered questions and challenges that need to be addressed in order to fully determine its validity. I encourage further exploration and testing of this idea, but with careful consideration and a critical approach.