On The Objectiveness of Mathematics


This article is written to describe and clarify some issues on mathematical philosophy. Be aware that this article is not yet complete, and is p00rly written. Alse cotains lota of mispellign and writin errörs.


Question: Why does some people think that math is subjective?

Because mankind does not always say 'what is really true', but says 'what should be true' and/or 'what they wanted it to be true' and/or 'what the others would like to hear to be true'.

This situation is somewhat related to current trends of thinking. For example, in a period when violence against animals increased and sensitive people exaggerated some thoughts in response, they tend to fall in the position of accepting or supporting all kinds of ideas that look as if defending animal rights. Humans find themselves in a situation like "Hmm, I must support this idea unconditionally." Even, in the past I was not surprised by the thought of animals as more valuable than humans as expressed by some 'worthies'. Of course the animals are very and very valuable, but no human should claim that animals are more valuable than humans.

That was just an example for better understanding. Current trends of thinking includes the idea of subjectivity and suspiciousness. Current trends suggest that:

  • Things that most people know are right, must be wrong.
  • I must go beyond the ordinary and surprize the people, because this is how smart people behaive.
  • Absolute knowledge is harmful, because 'suspected truth' always helps me to manipulate the other people's ideas.
  • When I cannot oppose in a discussion, I can argue that the presented argument cannot be confirmed with certainty and there is always the possibility that it is wrong.

I am not a psychologist, but I just wanted to share some of the my observations about this kind of reasoning. That kind of invalid reasoning leads people to accept fake truths about reality. Subjectivity of math is just another trend among the people who don't really understand the mathematical logic and philosophy. (You can even find some 'very smart' people involved in this trend, with PhD in Mathematics, Computer Science, or Philosophy.) Just like the other trends like 'the simulation argument', or 'the evolution by natural selection', or 'AI takes over the world' etc.

These kinds of trends are in form like \(cos\) or \(sin\) wave. I am sure that some of these trends will be absorbed eventually with the spread of the real truths and correct reasoning.


Question: Was mathematics is invented or discovered?

Mmathematics do not have a well defined definition. What we understand from the mathematics contains many different consepts such as notations, axioms, proofs, algorithms, theories etc. Some of these concepts are clearly human inventions such as notations and symbols, but many of the others are just discoveries of the mathematical reality.

Obviously, this article neither refers to representation of mathematical objects and properties, nor formalization or axiomatization of the mathematical consepts, but refers to the mathematical reality itself.

People investigates and discovers facts about this reality using man-made formal systems. The axiom and inference rules of these formal systems are determined by the human hand. Those formal systems can discover different proper subsets of the mathematical reality.


Question: Does there exists some mathematical system with a theorem \(2 \times 2=5\)? What does it mean if so? Does not the presence of different formal systems indicate the existence of a different mathematics?

Not just for this theorem, but for any theorem \(\phi\), there are infinitely many systems that can prove \(\phi\) or \(\neg \phi\). Even the bizarre thing is that, some of these systems can prove both \(\phi\) and \(\neg \phi\).

A formal system is either consistent or inconsistent. An inconsistent theory can prove every proposition, regardless of whether it is true or not. Because by definition there exists some formula \(\phi\) such that both \(\phi\) and \(\neg \phi\) can be derived from the axioms of the system. This contradiction eliminates the validity of the system, and such a system is completely meaningless obviously.

Similarly, a consistent system can be either sound or unsound. This time, an unsound system does not cause a syntactic contradiction, but it proves a false theorem that is contrary to the semantics of the system. That is, even though the formula \(\phi\) is not true, the system rules the trueness of \(\phi\).

Despite everything, if a sound and consistent mathematical system judges that \(2 \times 2=5\), then at least one of the following four cases has been occured:

  1. The number \(2\) no longer represents the twoness.
  2. The number \(5\) no longer represents the fiveness.
  3. The \(\times\) sign no longer represents the multiplication operation.
  4. The \(=\) sign no longer represents equality.

That is to say, the semantics of the system seem to be different from the semantics we are used to.(1) Whereas, there is one (and only one) truth over the same semantics.(2) The existence of different semantics does not imply the presence of different truths. A different mathematics could only be possible by arriving to contradictory results on the same semantics, but this is clearly against to the notion of the mathematics.

Trueness of \(\phi\) on theory \(T_1\) and trueness of \(\neg \phi\) on theory \(T_2\) can be expressed as \((T_1 \implies \phi) \land (T_2 \implies \neg \phi) \) on a new theory \(T_3\). Thus multiple semantics can be reduced to universal semantics, and that case does not make mathematics different, but implies the objectivity and singularity(3) of the mathematics.


Question: What are some possible corollaries of different axioms instead of current axioms?

Normally axioms are first determined in a formal system and then theorems are proved by following some mechanical rules. Indeed, this is historically not accurate for math. Mathematics has probably existed for thousands of years, but the first known axioms belong to the Euclid. Also, interpretation of the axioms at that time was very different from what is now understood. At that time, axioms were not developed to describe all mathematics, but they were used as the basis of some sub-theories and systems.

In fact, axiomatization of mathematics was made long after the mathematical developments had progressed sufficiently. Current foundations of the modern mathematics are based on ZFC set theory which developed in the early 1900s. People did not derive mathematical theorems from axioms. On the contrary, they obtained an inclusive and consistent set of axioms based on already known theorems, and mathematics itself seems to be different in this respect from any formal system.


Question: What if somebody prove that current axioms are inconsistent? What would happen? Would mathematicians quit his job?

Of course none of these would happen because of the reasons above. Mathematicians would just "patch" the axioms and resolve the inconsistency. Nobody would get hurt and there would be nothing changed in the world.


Question: If -counterintuitively- axioms are obtained from the results, where do the results come from?

There are two layers of mathematics in this universe.(4) First layer of the mathematics are present in the human mind, and the second layer is built on top of the first layer. The second layer of mathematics is highly syntactic and historically abstracted from semantics, and 'mathematics' is usually meant by this layer.

The soundness and consistency of both layers is independent of each other.(5) Human-created, or more accurately 'human-formalized' mathematics are indirect results of the mathematical reality which belongs to the human mind. Human mind has some implicit axioms either finite or infinite, such that they cannot be fully expressed in natural language, and probably no one even knows these axioms, but everyone agrees(6) with them without even noticing.


Question: Where do come from the axioms of the human mind?

They are a priori truth. They raise from the reality, and their proof is belief.


Notes

  1. The main point is not to draw attention to symbolic differences, but semantic differences.
  2. Undecidable propositions are out of topic in this manner. The idea of singular models can be expanded without loss of generality.
  3. Equivalence of singularity and eternity would be another topic.
  4. For outer universes, there are some more layers probably.
  5. Not completely independent for this manner: For two layers \(L_1\) and \(L_2\), \(Sound(L_2) \implies Sound(L_1)\), but the opposite is not always true.
  6. If they have sufficient intelligence and knowledge of course.