Are the laws of mathematics and logic, such as if a=b, and b=c, then a=c just constructs of the human mind, or does the universe hold an innate logical structure to it, which the physical part of the universe must abide by?
The answer is: we don't know. As the user Seldom indicated, Do numbers exist independently from observers? is a very related question.
What we do know is that there seems to be an incredible correlation between logic (and mathematics, for that matter) as we see it in human mind and the way the universe behaves. For some reason, everytime I make some logical conclusion on my mind or even a calculation, "reality" seems to agree with it. And everytime we discovered a discrepancy between our logic and reality, it was because we had erroneus models/premises.
So until now we have so much success that we induce that the universe seems to work with some kind of correlation with logic. Of course, there are some possible reasons that happens:
- Logic is a structure of reality
- Logic is an abstraction made by humans looking at reality
- Logic is a human way to look and filter reality
Again, very similar to the numbers question.
Insofar as any of these relationships are defined and scoped by the human mind, so are they contingent upon the human mind.
Dependency/independency of some mind begs all the wrong questions and results in meaningful and senseless statements like "equality exists outside of space and time" and "laws of physics, equally, are part of the universe, regardless of human observation." These, in turn, result in all kinds of sloppy language.
This is not a meaningful question in logical positivism. The question of whether something "exists" or "does not exist", especially in relation to the existence of other inferred things, like human or animal consciousness, is an abuse of language in the sense of Carnap.
So this question has no answer, and it needs no answer, because it is not a real question. It's just a bunch of words you put together which fool your brain into thinking they have meaning when they don't.
Basically note that we are living in a world that is certainly not self-contradictory and has rules for itself (some approximately discovered and some not even in any approximate way, maybe), but surely there should be some axioms for it to be based upon, as we can easily say something is possible (according to such axioms) and something is not possible (again according to that axioms) and such sentences are all meaningful (although there is aways the possibility that the one who says these expressions be wrong in respect to his objectives, so I am talking about the ontic knowledge and not epistemic, what really is and not what we think to know of reality). Thus, logic by its own is beyond the human kind, even humans exist based on the same logic that other things exist, they all belong to the set of possible objects in this world according to its underlying axioms, or say logic rule.
However, mathematics is perhaps the only branch of science which has not its base on witnessing the outside real world, you can imagine a set of axioms and logical rules to create your own self-consistent mathematical world, and substituting some of the axioms or logical rules by their antonym you may still be able to create another self-consistent mathematical world for your self and yet be happy with your creation. None of these logical rules may also have no counterpart in the outside world and that's it, logic exist in human minds and outside it!
I believe that the laws of mathematics are independant of thought. The names, symbols, etc that we assign to numbers are irrelevant, but the ratio between the radius and circumference of a circle, for example, are constant whether we observe them or not.
The laws of physics, equally, are part of the universe, regardless of human observation. We spend time and effort trying to figure out what they are, not inventing them.
I would like to believe, also, that the laws of logic are similarly imutable. The way we express them make it feel like they are human inventions, but I believe they are constant facts of reality that we discover and decide on conventions to express, rather than creating and enforcing rules on reality.
We would not observe water flowing uphill (observe, not just perceive), or 1 individual object coming together with another and forming a group of 3. However, when it comes to logic, it is possible to perceive actions which seem to be founded on a belief that defies logic, making it easier to think that it is human-created.
logic is a tool used by beings capable of reasoning, we use this tool to create valid reasoning and to understand valid reasoning, so when a human observes the universe he uses this tool to create a valid reasoning to explain what he is seeing, math is the lenguage we use to understand how natural phenomenons work, natural phenomenom exists with or with out humans the way natural phenomenoms work is the same with or with out humans, mathematics is the language we use to translate the way this phenomenomswork into human understanding, so logic and math are correlated but they only exist when theres is a mind to use them.
It's not like you can take a telescope or microscope and "see" logic. The "laws" of logic don't mean anything except inside a mind. Just as the "laws" of physics are only meaningful inside a mind (brain). They represent our understanding.
I think logic is not a "thing", its a process and activity of a mind/brain. It's like saying "seeing" exists without eyes or "running" exists without legs. Something has to be "doing" the logic.
Let's start with computers. What do computers do, if not logic?
I decided that they don't, and here is my argument, based on a definition of logic: logic is a form of thinking that attempts to solve a problem by manipulating meaningful symbols via formal rules and processes, to get a meaningful result.
So, whatever is happening in a computer, it's not really logic until someone has used it to solve a problem and/or assigned meaning to the symbols being manipulated by the logical mechanism. And a computer can't do that part of logic without a human (yet).
Consider the abacus. It can't do math until you have assigned a meaning (ie, number) (in your head) to each of the beads being moved. Without a human to interpret their meaning (ie, "read" the results), it's just a bunch of clay beads moving around on a set of rods. The same thing can be said of Babbage's calculating machine.
And it's the same with computers. Until the results of the (much more complex) electronic manipulations in the computer is interpreted or assigned meaning, it's not really doing logic, or solving a problem. Without a human to interpret (assign meaning to) the results, it's just a bunch of electrons moving around.
For example, an open circuit could just as easily mean "true" as "false" The meaning is assigned as a convention, and the rest of the computer is built based on those conventions. ASCII character 45 could just as easily mean "Z" as the letter "A".
In addition, there is the issue of defining the problem that is to be solved. This is another step in performing logic that computer's can't currently do. They have to be programmed to solve specific (human) problems.
Yes, they seem to have done a lot of thinking for you, but the problem the computer is solving was defined and programmed into the computer by a human, and the results of all that computer processing are meaningless without a human to read or interpret it (ie, consume the output), so I say computers are not really performing logical thinking.
In the future, computers may be able to be self-aware, emergently conscious, capable of reflection, progamming themselves, and assigning meaning to their own results. In the meantime, they are an extension of, and a tool for, human thinking. They have been described as our "external brains".
When humans read, they are assigning/giving meaning to symbols on paper or some other media. When humans think, they are manipulating mental symbols (concepts). Logic (and math) is the manipulation of mental symbols following formalized rules, in order to to solve a problem.
So what do I mean by "meaning"? Meaning is context. Concepts are only meaningful in the context of simpler concepts they are built up from. That is how sophisticated ideas are constructed. That is how we learn, by gradually increasing the complexity of our concepts.
Some possible implications: Logic and Math are not something "out there" that we "discover". The rules of logic and math are "constructed" by us, as our mental processes becoming more and more abstract and formalized. This is why logical thinking is hard, and takes practice.
I also agree with the "embodied mind" theorists that many if not most of our simpler concepts are understandable and can only come from the experience (ie, context) of living in physical bodies. For example, the concept of "color" is not meaningful without color vision, and a brain that is hard-wired to interpret it for us.
Nevertheless, I do think that logic and math are saying something universally "true" and non-relative to our human condition, in the sense that I think that if aliens were found, they would have developed logic and mathematically systems that are very similar, if not identical to our own (thanks goes to Carl Sagan for that idea). The more abstract our math and logic become, the more universally "true" they become. But, that truth cannot exist outside of a thinking mind. It's not "out there" to be discovered, it's "in here" and constructed, by hard thinking.
Of course, what is amazing, and almost mystical, about logic and math is how well things "in here" allow us to predict and accurately manipulate things "out there". That reality can be abstracted, and those abstractions re-applied back to real things is amazing. As Einstein once said "The eternal mystery of the world is its comprehensibility."