Is Maths based on truth?

Is math discovered or invented?

Ref: https://medium.com/@vovakuzmenkov/poincares-creativity-8e31d5031529

Most of us perceive a math as pure logic, where any statement is a ground truth, and a mere continuation of a logic construct. Is it so?

True inventors and enthusiasts on the verge of the cutting edge in their field, know there is more to it, and the reality is so much different from what people learn at school.

Important takeways:

• Poincaré concluded that geometry axioms are conventions.
  • To address the issue of mathematical truth, Poincaré suggested we first examine the nature of geometric axioms.
    • Do they exist as fixed parts of our consciousness, independent of and uncreated by experience? Poincaré disagreed. If they did, they would be so compelling that we couldn’t imagine or develop any alternative theories, like non-Euclidean geometry.
    • Should we then think of geometry axioms as experimental truths? Poincaré didn’t agree with that either. If they were, they would constantly change with new data, which seemed contrary to the nature of geometry.
    • Our choice among these conventions is influenced by experimental facts but remains free, as long as it avoids contradictions.
• Geometry isn’t about truth; it’s about usefulness.
  • He then asked whether Euclidean or Riemannian geometry is true and answered that the question is meaningless.
• Our concepts of space and time are also definitions chosen for their convenience in handling facts. Our scientists have limited time to observe everything.
  • With an infinite number of possible observations, an unselective approach to facts is no more likely to produce science than a monkey at a typewriter is to produce the Lord’s Prayer.
  • Poincaré argued that if a scientist had infinite time, simply observing everything might suffice. However, since time is limited and incorrect observations are worse than none, it is essential for scientists to make deliberate choices.
• Just facts alone is not science
• The more general a fact is, the more valuable it becomes.
  • There is a hierarchy of facts.
  • Facts that are applicable in many situations are preferred over those with limited applicability. To identify valuable facts, choose those that appear simple.
• By exploring extreme cases—either very far in space or time—we may discover that our usual rules are overturned.
• A scientist does not randomly choose the facts to observe. Instead, a scientist aims to distill extensive experience and thought into a concise form.
  • This is why a small physics book encapsulates vast amounts of past experiences and countless potential experiences with outcomes already known.
• The true work of an inventor involves selecting from a vast number of combinations to eliminate the useless ones or to avoid having to try them.
  • The rules guiding this choice are very subtle and difficult to articulate; they are felt rather than explicitly defined. Poincaré hypothesized that this selection process is managed by what he called the “subliminal” self.
• Mathematical solutions are chosen based on “mathematical beauty,” such as the harmony of numbers, forms, and geometric elegance.
  • Poincaré described this as a genuine aesthetic sense familiar to mathematicians but often overlooked or misunderstood by others. It is this sense of harmony and beauty that is central to the process.
• It’s not the facts themselves but the relationships between them that create the universal harmony, which he considered the only objective reality.
• What ensures the objectivity of our world is that it is shared with other thinking beings. This shared harmony and quality provide the foundation for the only reality we can ever truly know.
  • Through our interactions with others, we receive well-structured reasonings that we recognise as not originating from us but as the work of other rational beings like ourselves.
  • Because these reasonings align with the world of our sensations, we infer that these other rational beings must have perceived the same reality we do, thus confirming that our experiences are not mere dreams.
• When Quality is introduced as a third metaphysical entity, the preselection of facts is not merely subjective. Instead, it is grounded in Quality, which represents the essence of reality itself.
  • According to Phaedrus’ metaphysics, the harmony Poincaré referred to is not subjective. It precedes and underlies subjects and objects, acting as the force that counters capriciousness.
  • It is the fundamental ordering principle of all scientific and mathematical thought, essential for coherent thought and for integrating the separate languages of Science and Art into a unified structure.