What does mathematics rest on, if not human intutions about logical necessity and what follows from what?
Are human intuitions always wrong?
I submit that many animals of lower intelligence employ intuitive rudimentary mathematics. How is it that a Lemur can count as fast as a human and in some instants faster than humans? It's been tested and confirmed. What does a Lemur know about mathematics?
All of nature functions in a mathematical manner and evolution selects for those mathematical abilities which give the organism a survival advantage.
The petal growth of the lowly Daisy is a marvel of employing the mathematical efficiency of light collection found in the Fibonacci Sequence.
All patterns in nature have underlying universal mathematical values and functions. How is it that a chameleon is able to triangulate the precise distance to a bug sitting a body length away and unerringly flick its tongue to snag the morsel? Ever watched an octopus solve a completely alien mathematical problem, such as "unscrewing" a child proof lid from a jar.
We even made a Higgs boson appear from the Higgs field. Intuition or predictive mathematical precision? We build the most expensive machine ever to simulate the condition under which the Higgs boson would become manifest. And it worked!
Human intuition (cognition) leads to Theory and subsequent proofs confirm or disprove the accuracy of the cognitive intuition. Practical application of correct mathematical equations allow for human technology.
E = Mc^2 is a mathematical proof of a fundamental universal deterministic function, no?
Are there any known exceptions to that profound equation?
The Fibonacci Sequence is a universal deterministic mathematical pattern. The Second law of Thermo Dynamics identifies a mathematical deterministic constant. These are inescapable deterministic universal phenomenon.
Universal patterns are deterministic when they self-assemble into consistent repeating configurations.
Can you provide a non-deterministic example of a universal mathematical constant?
I take the word of cosmologists who to a person admit that when they test a theory, it confirms that the mathematical values and functions already existed in a generic universal form long before they tested their symbolic mathematics against these universal values and functions.
Our numbers and operators are just a convenient language to symbolize the generic universal mathematical values and functions, how they interact to form constant equations which allow us to predict a natural phenomenon. Our maths do not shape universal values and functions. Universal values and functions shape our maths!
If you cannot accept that then why do we employ mathematics at all? We are constantly discovering that no matter where we look we see deterministic mathematical values and functions. All of Science is founded on these natural universal deterministic mathematical phenomena.
Universal mathematics are staring us in the face wherever we look. They take an inescapable functional part in repeating patterns of universal values and functions. All our science is founded on observation of universal generic mathematics. Why deny this incredible evolved cognitive ability? We're not alone in this.
The logic and precision of the table of elements is a beautiful example of universal mathematical expression in fundamental physical atomic patterns. We just represented it with human symbolic mathematics because we can and it works!