Math is
I think basic foundations of math are invented, the world outside of human pattern recognition and abstractions doesn't seem, to me, to inherently have even the minimum basics of mathematics. Higher-level math are discovered within the framework and constraints imposed by the foundations.
As for algorithms, some seem to take shape because of prior human-made constructed constraints, changing the constraints changes the optimal algorithms, but some seem to me inherent in the world itself, I'm not sure. Either way seems to be discovered.
High-level computer programs are invented I think.
or I thought.
Now I sense an inconsistency in my point of view, if high-level programs are invented how come algorithms are discovered. I can't pin a fundamental difference in nature between an algorithm and a high-level program. Assuming High-level program here means the "design of a High-level program" and not the concrete implementation.
Maybe this is a false dichotomy of sorts, or the definitions in my brain are wrong somehow. I took math to be invented, yet I consider it the result of human pattern recognition. What's the difference between "pattern recognition" and "discovery".
Maybe invented and discovered are kinda the same thing. Maybe the better answer to all three is, math, algorithms, and high-level programs are "pattern-recognized & abstracted into existence" by humans, little by little and layer by layer.
I don't know. I also don't claim consistency in my rationals :] This probably requires more precise definitions of terms and more a nuanced approach. I haven't thought about this a lot before, so my answers here are more like gut reactions.
These questions sound like a good way to start some heated arguments with people over dinner lol
@rozenglass @sir math is neither invented nor discovered, it is only taught. If you are sitting in the corner brainstorming that's not math. Math starts when you start explaining it to another person. You don't need arithmetic or even a concept measurement to build a tent, you need them to tell me how to build a similar tent on a small clay tablet, cause they heavy af.
I'm just saying none of that is math. Math is not the science of some fundamental truth (in a way it proved that such truth is impossible to define https://en.wikipedia.org/wiki/Tarski's_undefinability_theorem). I'm also not saying that invention, discovery or common sense do not exist. They do. They come from various places, be that the physical world, or your imaginary unicorn land. Still not math.
I'm playing around with some sticks and woah a tent appears! (not math)
Can I do this again? Woah I did it again! (still not math)
This is kinda tedious, maybe I could get Bobalice (ancient Babylonian name) to do this for me. How do I explain it though? (this is where it starts)
Bobalice after explanation: I knew this before you were born you dumbo (math still happened)
> The theorem applies more generally to any sufficiently strong formal system, showing that truth in the standard model of the system cannot be defined within the system.
Nice Wikipedia link, thanks :)
> I'm playing around with some sticks and woah a tent appears!
I'm not sure if this tent story is an analogy or meant literally, if it's an analogy to the discovery/invention of math, then no, I think math can exist with one person only, and without the communication of information between multiple people. But it might be that my definition of math, and yours, are not exactly the same.
But, if the tent story is meant literally, then I don't see what math you're talking about that was introduced when I explained to Bobalice, who knew already how to make a tent before I too made it independently :]
> I'm playing around with some sticks and woah a tent appears!
As a side note which is probably irrelevant, but I'm not sure, it might be that you designed the tent with way more intent than this. i.e. not accidentally, but because you thought something like "I need shelter from the sun, so I need a barrier, and I need something to hold up the barrier, nope, I need at least four of them to keep the balance, and I need..." etc.
Tent is an analogy of any discovery or invention or knowledge. The rigorous explanation is math, even if it's an explanation of something well known. It's useless you would say, and sure it is, until you are sufficiently advanced to build a machine (in the broadest sense) that can make tents, then it becomes your holy grail.
Something that exists in just one person's head will not exist for long. Unless of course they make it into a machine which no one else understands. I for one welcome our rob... no I don't!
Math is the teachings of our kind. I can't imagine any other definition that wouldn't be religious.
I see.
> Math is the teachings of our kind. I can't imagine any other definition that wouldn't be religious.
I think I agree about math. On a side note though, I'm not sure about basic physics that seem inherent in the world. I'm an atheist and I still can't imagine an explanation to them that wouldn't be religious. My imagination ain't the best though.
> I for one welcome our rob... no I don't!
oh noes! the robos will take our math :[
ah well, the robos are the making of our own hands, our children in a way. I for one welcome the creations of our thoughts, they might be our legacy, and we will always be the fathers, however weak and flimsy we would be in comparison to the binary mathematical savants.
> build a machine (in the broadest sense) that can make tents, then it becomes your holy grail.
my holy grail would be a machine that makes good code
or good food
:P
Our good robot servants we understand, the one we don't/can't understand will make us its servants, or worse deprecate us. That's what I meant. If it's not that machine, and is just coincidentally unexplained, than someone will come up with an explanation eventually. And until that point we'll be wondering "is this math or is this our new overlords?" with some thick plot, much better than terminator or matrix.
@rozenglass @namark your argument from physics is a stretch. The physical universe follows behaviors which can be modeled accurately with math.
Perhaps. I'm just trying to start from the lowest most basic principles possible, they usually, incidentally, tend to be physics most of the time. The core of my argument though, lies in discussion of human pattern recognition, and not formal physics.
As for modeling physics with math, we do so in a manner accurate _enough_ for most of our current purposes, relatively consistent with our own perceptions and interactions with the world, and usually by hammering the physics away till it's "good enough", ignoring errors and loop holes, rounding values, and creating higher-level abstractions. But we have no way of knowing how close it is to an actual representation of reality and the underlying mechanics of the world.
In my view, it all becomes really weird when you start looking at physical constants, like the Planck length, or the speed of light.
Maybe there's a lost distinction here between Physics as a human field of knowledge continuously developed, and physics as basis for our world. The first is an attempt at reserve engineering the second, and striving to build a wholesome consistent theory that describes it, a complete abstraction. A similar distinction could be made of math perhaps.
Both could be viewed as leaky abstractions of sorts, and again I iterate, I think pattern recognition (discovery) and building abstractions (invention) are probably the same brain-function, the moment you recognize something, is the moment you build an abstraction for it.
Anyways, I'm out of my depth here, and this is all an attempt of on-the-fly reasoning, it probably has more holes than this post has characters, and this post has quite a few characters.
Well, I'm kinda talking on a much lower level. My point is that we select two groups of atoms in the endless sea of atoms, and then call them the "first apple" and the "second apple". There's nothing inherent to the universe that mandates this group of atoms be called an apple, what calls it an apple is us, humans, and only through abstracting matter into a single unit can we start counting. Counting doesn't exist in the endless sea of matter and energy outside of human perception. So, I call it an invention just by virtue of it being the result of made up things in our brain. But again, a better discussion around this calls for better definition of terms, agreement on what "invented" or "discovered" mean.
If we raise the level of discussion above the point at which man recognizes a pattern of matter as a "unit", then I think man can discern that two apples are more than one, and that putting them together makes them even more. I don't think that this is taught, but it is discovered within the framework of the fundamental basic inventions. Very young babies seem to me to have some form of recognition of count, simply by seeing one crying when one of two balls they were playing with was taken away.
Again, definitions of terms.
I intuit, that in this context, invention and discovery might be considered the same thing, You "discover a pattern" when you recognize it, yet you also at the same time "invent an abstraction". In our brains, the act of discovering patterns is, itself, the act of inventing abstractions. The moment you recognized the shared properties of a group of atoms as being coherent and similar, while being different from different groups of atoms, is the same moment you invented an abstraction that labels it as a unit; as "something".
Buuuuut whatever, the definitions of all terms mentioned in those questions are flexible enough to make any answer justifiable at different levels of analysis, and I can probably waste years philosophizing about them, writing useless books and text walls, and talking people's ears out, so I'm just gonna stop now :P