Manipankaj9@gmail.com
I humbly think the way mathematics is understood conventionally requires deep shift. Infact I have already tried to resolve some of the foundational paradoxes in mathematics by making mathematics time human and relativistic rather the assumption of absolute independent of human. I think Time ,Relativity ,Human do have essential role in mathematics evolution over time, though it’s quite slow compared to physics . Infact for the application in finance, economics ,human behavioral, new mathematics is being developed because the fundamental assumptions of these conventional mathematics like calculus, statistics etc. are practically fundamentally incompatible in modelling human world. Conventional Mathematics seems to be mostly in classical world but human behavior is sort of quantum world.
But before that, for the entire philosophical discussion of human and mathematics including of Platonism and Non Platonism etc. I think one needs to understand first what is Human mind /Consciousness first i.e. The Observer.
Without understanding the observer, the argument about the observed is seemingly incomplete !
Even the Platonist thoughts like Mathematical Reality existing independent of Human was also produced by Plato’s biological human mind only !
What is is meant by the word “independent” is also very ambiguous at times.Platonic approach seems Paradoxical in itself while in his Theory of forms, forms are not well defined, well acknowledged by himself.
Platonic approach : On one hand he talked that Mathematical Realities exist beyond Physical Tangible world in the abstract world independent of human but can be accessed by only human mind intellect. I mean how can something independent of human can be accessed and understood well by human mind itself unless the both are deeply interlinked and dependent on each other ! For that one needs to understand first human mind itself. That’s why the essential role of Consciousness. Without understanding that in detail humankind will always be arguing and fighting over these opposing thoughts which is true. Mathematics may superficially appear independent of one specific human but not collective humans.
On one hand it talks about mathematical reality beyond physical materialistic world and on the other hand Platonists try to explain the physical materialistic world based on that only !
I personally see so far it’s Paradoxical . Mathematics could be Independent and dependent both how an observer’s mind looks at it !
I have also tried to show how Mathematics is also Time and human Dependent to try to resolve foundational paradoxes and it gets significantly.
When these things are understood deeply one can possibly realize that Mathematical Realities may not be independent of human mind. Infact while dealing at the scale of Infinity like in Infinite series and all, the role of human becomes quite significant how one tweaks the rule.
Mathematics has many components some seemingly natural like Pythagoras theorem , some humanly created like numbers etc.. Even Naturally existing realities like Pythagoras theorem etc are based on intrinsic physicalities, symmetrical aspects. I see mathematical realities like geometry too something following the fundamental characteristics of Physical Nature /Universe we live in. It’s mutually both ways. That’s why many features of physical and human behavioral world like randomness, entropy, symmetricity, asymmetricity etc. exist even in mathematical realities for example Numbers their distribution etc..say Prime Number Distribution and their behavior etc..
These empirical evidences strongly seems to convince me that natural mathematical realities, physical realities , human realities all are interlinked with each other..The difference between all of them seems to be due to illusion of consciousness. After all from where the logic behind the fundamental formal axioms that come that define mathematical systems, they come from human physical world analysis.. and then the same mathematics is applied to study physics ! So it’s like one type of physics trying to explain another physics !
But yes all Mathematics is not Naturally existing ,some areas are humanly created as well like complex numbers. And even Randomly created mathematical system will also have some inherent patterns because the logic and principles on which human define new mathematical are also based on physical realities experience and human behavior etc.
The very concept of Numbers etc. I do believe that exists on Human Mind. When someone asks how many Natural Numbers exist...My answer would be rather how much one can count up to or imagine upto rather than absolute infinite. This seemingly Platonist belief that Infinite number exists is possibly the root cause . Mathematicians need to first understand human mind in my humble view.
And at the deeper level when we study the properties of Numbers ,we do see their behavioral similarity to human mind neurological behavior and also physical world behavior ..For example there exist properties like mean reversion, randomness, entropy etc. even in mathematical numbers that can be found everywhere including human and physical world. Even the geometrical structure of human is mathematical.. How Ramanujan's mind or intuition in the mind know mathematical discovery if they are independent. How does mind establish truth or not about mathematics if they are independent. How does mind know the external truth about mathematical or physical realities if they are independent?
At the same time, there lies fundamental friction between Mathematics and physics which is also one of the foundational causes of renormalization type issue though it’s another matter than it has become useful just like complex numbers which was created to hide Inconsistency in the algebra. This is also very important issue to understand that even if one randomly creates a mathematical system and tries to calibrate into real world situation through physical experiment , one may always get some predictions correct out of many , but that doesn’t guarantee its validity in principle. This forceful calibration technique in finance and economics has even bankrupted many large funds globally including top class banks. The solution is possibly deeper than what we think.
The problem is to check the fundamental incompatibility in the existing mathematical system that is like a framework to model physics. Mathematics has to be made real world especially in the context of managing infinity. Like Real Numbers ..can be infinitely divisible but in physical world and atom may not be infinitely divisible. Or say mathematical theory of Set believe in simultaneous existence of its elements but Simultaneity in Relativistic Physics is different. So, over the time Physics has changed from classical to quantum etc..but Mathematicians still prefer to remain in Platonic timeless existence due to strong mental inertia or say classical world .
As in Feynman diagram or otherwise why charge of an electron or mass should depend superficially on the inconsistent mathematical measurement having numeric value. Numbers can always be manipulate by choosing different reference points. At closer to 0 distance Mathematically it becomes infinite based on arithmetic rules but physically it is not Infinite. The point is for real world application mathematics as the frame needs to be made real world by redefinition rather than applying the same inconsistent system. As that’s done it would suffice the purpose for sometime but this fundamental issue would always pop up in future unless the fundamental system is changed and revised how physics is done and how mathematics can be made real world. That’s the reason I am skeptic about Renormalization etc. because fundamentally the very concept of Infinity is dubiously understood in both mathematics and physics!
If one goes deeply one can find the dubiously Uncertainty in mathematics itself while dealing with Infinity like Ramanujan Summation, Abel Summation also linked with the celebrated Riemann Zeta function ; These Infinite convergence methods are also applied in Renormalization techniques e.g. Casimir effect.etc in QFT. So many Non Renormalized theories are discarded just because of such issues . So I see like in finance and economics a mathematical system is first defined and then calibrated with physical results in biased ways to fit and then claimed to be working perfectly ! That’s the reason mathematical models in economics finance remain badly performing. So, as Renormalization has come a long way since Wilson time, one should continue that but I humbly believe there is need to look at the foundation as well so that future discrepancy/Inconsistency doesn’t occur in some other unexpected forms.
Mathematics itself has its own intrinsic physics. It also has randomness, entropy etc That’s different from the normal real-world tangible physics subject. Mathematics itself is foundationally incomplete or Inconsistent in Godel Incompleteness context so when reference frame itself has such issues how far physical world realities modelled on that can be consistent or complete. I doubt. This has been raised by even likes of Stephen Hawking and others. Even some randomly created mathematics may be forcefully calibrated to fit in the reality.
Bertrand Russell told : Physics is Mathematical not because we know too much. Infact we know so little that only its mathematical properties we can discover.
So I don’t seem to see the application of Mathematics Science some what mysterious rather obvious as far as they are able to do, of course there are many unsuccessful applications too.
In a nutshell I humbly believe there is serious need to study and understand the philosophy where science and mathematics and others are going because in this current technological age the Philosophy of Artificial Intelligence etc . also seems to be misguided significantly trying to mimic human using mathematics without understanding mathematics and human mind deeply!
I humbly think we need to come out of mechanical system where Philosophy is treated in bad manner .,Infact as I read somewhere in Ph.D. (Doctor of Philosophy) the essential component of Philosophy seems to be missing these days and the geodesic of science, mathematics etc. Seem to be misguided affecting the day to day society.
