Reflection on Basic Mathematics I — Are Mathematics real?
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
Gödel said that mathematics how they are formulated now contradict themselves. Yet reframing the problem we would say mathematics is not well-defined. That frame makes it much easier to solve for contradictions than before. Then lense through which we should see math is less logically but more through sets and especially functions.
Introduction
In this article series, I want to reflect on certain basics of mathematics, to first finally pass the 1st semester course, and next to be able to thoroughly check my theorems. It is also my homework, though. Also, the text will change a bit down the road. What a great professor I am having this semester. I am sure you can do it better than I can, yet you push me to do it. Thank you!
The trick is that I do not have objective feedback except from some audience on the internet. That means that I am probably alone with this, except from the feedback from fellow students, which, I hope, will come this semester. Think freely!
The eroding math we learn, see and practice is especially the problem. The math at our university is broken. And this issue is really awful. As our science is loosing its foundation. But it can be solved with real math. And I will argue to you why.
At the university, we are able scientists, thinkers, and creators. It does not matter if you are in the first semester or an established professor. We seek the truth. Or we should do that. And this is hard work.
This is also why I left early the other day. I am staying longer every day until I find a reasonable amount that fits my lifestyle. I needed time to make all of this conscious for me. I am sorry. It would have been nice to talk to a beautiful girl, or enjoy some nice time with my fellow students.
Anyhow, once we can forget about our egos and start to seek it again, we will gain back our freedom. And I know this won’t be easy.
Your victory is merely a decision, though you have to make it constantly. Every day. Or this is how it is for me.
The pleasant thing about math is that you can build a building if you are just careful. If you find a faulty stone, you just replace it with a good stone. Easy. Well, maybe. Occasionally.
Gödel
At first glance, even math seems lost. There are two theorems from Kurt Gödel that some take as the justification to not seek the truth and stretch it however they please.
„Jedes hinreichend mächtige, rekursiv aufzählbare formale System ist entweder widersprüchlich oder unvollständig.“
"Any sufficiently powerful, recursively enumerable formal system is either contradictory or incomplete."
So, some people abandon math as something that is not getting them closer to the truth, so they are allowed to torture, rape, and abandon math altogether.
But Gödel said nothing about these things. In fact, there are complete mathematical formulations. For example, there is a complete axiomatic system for the Euclidean Geometry. Which makes sense because we can literally build monuments like world wonders based on these!
But Gödel did not stop here either. He said:
„Jedes hinreichend mächtige konsistente formale System kann die eigene Konsistenz nicht beweisen.“
“Any sufficiently powerful consistent formal system cannot prove its own consistency.”
So if I can’t prove the truth it does not exist, right? Well, let’s try to approach this problem like a mathematician. I reformulate the problem articulated above by my imagined critique:
If we can’t prove that some systems are completely consistent within themselves, can we then conclude mathematics is not real?
Think about it, again, and read the new formulation of the problem and compare it to the theorem. The theorem says nothing about a thing, that there are no true axiomatic systems at all. Thus, we can find true axiomatic systems.
And there you arrive at the mathematician's work and craft. We need to build a building with solid stones and if we identify a faulty stone, we replace it with a better stone, such that the building will hold.
And often, we just say the things differently or in another language (this can be different math, different natural languages, or maybe also different programming languages or patterns) and see if the new formulation still causes the same problems.
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Conclusion
So, by just reframing the problem, we can conclude mathematics is real and this is also what Kurt Gödel thought because he was Platonic. Being a Platonic means that you acknowledge your perceived reality is a manifestation of ideals (objects) from another (higher or more mighty) realm (or space). Just as you can map different spaces to each other with objects, we call functions.
And that’s it for the first article. More interesting thoughts on mathematical structures follow in the second article.
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