Numbers and the mathematical operations (like addition) that manipulate them have no physical reality. They are not made of stuff with any structure or substance. You cannot detect numbers / operators with your five senses, even when augmented with advanced technologies. Just seeing or hearing a number gives it no more reality than seeing or hearing about Wednesday or beauty gives those concepts independent reality. But unlike beauty and Wednesday, numbers seem to have more than a passing connection to what we call really real reality. Really real reality can be translated into numbers and equations. Numbers are the language that really real reality is expressed in.
Yet, numbers, mathematics and the associated operations like addition, subtraction, etc. are mental concepts. To the best of our knowledge, they are solely human concepts. Numbers, mathematics and the associated operations have no independent structure; they have no independent substance. They are akin to other mental concepts like units of measurement, the idea of beauty, the concept of Wednesday, human emotions, and so on. In a cosmos devoid of life, concepts like hours, ounces, miles, Wednesday, love, depression, etc. have no meaning or foundation. Can one electron love another electron? Do electrons have the concept of days and time off? Can an electron give a lecture on the quadratic equation to a collection of other electrons? The number two or the quadratic equation doesn’t exist in a cosmos devoid of life, unlike say a star or even an electron. Now all of that is assuming that a cosmos devoid of life is devoid of intelligence. But if an intelligence created the cosmos – like a computer / software programmer – then that cosmos has all of the mental concepts contained within the intelligence that created (programmed) that cosmos.
Now let’s say that human intelligence is the only cosmic intelligence and that human intellect has therefore invented mathematics since there is no even higher intelligence that has infused the cosmos with mathematics waiting to be discovered. My basic point here is that if humans invent something, if humans make the rules, then humans can change the rules. Before humans existed, the concept of 6 x 7 = 42 didn’t have any meaning. Electrons have no concept of 7 or of 6 or of multiplication or of equality or of 42. Humans gave all that meaning. But then humans could change their own rules and declare that 6 x 7 = 24. If everyone agrees that 6 x 7 = 24, then 6 x 7 = 24! There’s no higher authority to say otherwise.
So here’s the concept of reality in a lifeless Cosmos.
Wednesday doesn’t exist. Anger doesn’t exist. There is no Merry Christmas or Happy New Year. The concept of days off doesn’t yet have existence. Ditto the notions of soap opera, horse opera and space opera. The contradictory concept of married bachelors hadn’t yet reared its head. Nothing in existence in a lifeless Universe had any concept of sleep and dreams.
Daylight Savings had no credibility as a concept immediately post Big Bang; the concept of pizza would take another 13.7 billion years to arise; language of any kind was absent; and while physical processes were afoot, the concept of physics didn’t exist.
The idea of an automobile has yet to arise. Since we have automobiles now, did that concept exist before life existed? I think not.
Music didn’t exist, yet music is totally a mathematical construct.
Then there is the concept of red. The wavelength(s) exists, but does redness exist if there is no life-form that can perceive red? This all seems to be a mirror of that old philosophical puzzle – if a tree falls in a forest and there is nobody there, does it make a sound?
Now if none of these concepts had any reality in the era before life arose, what makes anyone think that mathematics, numbers and their operations did? In the absence of life, 2 + 3 = 5 is still logical, but only if the concept of two and three and five and equals and addition were somehow ‘out there’. So the real question is, if all of the above concepts had no meaning in a lifeless Universe, a Cosmos devoid of life, then why should mathematics be any exception.
But okay, if you say that 6 x 7 = 42 even in a lifeless Cosmos or even in a Cosmos devoid of anything and everything, then you’re in excellent company. It’s hard to argue that a row of six things repeated seven times over will not give forty-two things regardless of what language you call “six” or “seven” or “forty-two” in. Now, my next thought is does “E = mc-squared” exist in and of itself in a lifeless Cosmos? In other words, get stars convey mass to energy in a Cosmos devoid of life? If you answer “yes” then presumably all mathematical equations that describe the laws, principles and relationships of physics exist independently of life.
For example, if gravity existed pre-life then both Newton’s law of gravity and Einstein’s slight refinement of same existed pre-life. If electromagnetism existed pre-life, then presumably Maxwell’s equations also had some sort of structure and substance pre-life. If planets orbited the Sun and the Sun orbited the galaxy and the Moon orbited the Earth before there was even a humble proto-cell to appreciate all of this, then presumably Kepler’s Laws of planetary motion also existed.
The follow-up question then arises is how can all of this mathematical regularity, mathematical certainty, mathematical predictability, arise? “E=mc-squared” seems both designed (there’s only one possible equation that can relate the relationship between mass and energy) and fine-tuned (for any value of m, there’s only one possible value for E). To generalize, if all of the equations that govern the laws, principles and relationships of physics, hence rule the cosmic roost, existed pre-life and all of these equations are designed to do the job they do and give fine-tuned answers, then some sort of explanation is required.
Can it all possibly be by chance (the Mother Nature Hypothesis) or does there have to be a designer behind the Cosmos who is rather fond of maths? And if the latter, is the designer a supernatural deity or a computer / software programmer? If the latter, then we ‘exist’ in a simulated landscape that’s mathematically determined and follows mathematical rules, like 6 x 7 = 42. 6 x 7 = 42 exists independently of life because it was so constructed that way by… ?
By the by, FYI, there’s a good NOVA documentary on YouTube that partly covers the ‘invented vs. discovered’ issue called “Mystery Maths In Universe Science Discovery Documentary 2015”, and also found by plugging in “The Great Math Mystery”.