That's more a chart of dependencies, and only partially at that. Mathematics uses logic as a tool, but essentially everything we do in mathematics was motivated by something else. A lot of things in math started out as heuristic concepts because people felt that it should be so, despite not being able to give a good logical foundation for it. The complex numbers and the calculus were used long before their logical status was clarified. Heaviside's operational calculus worked extremely well. As far as I know, nobody has been able to create a rigorous version of operational calculus that was as dynamic as Heaviside's, but such attempts have given us wonderful tools like Laplace Transforms and Generalized Function theory.
And even in physics a lot of things happen that would make mathematicians cry, much less logicians. We differentiate and integrate functions without establishing that this can actually be done, because we're pretty darn sure that shit like Dirichlet's function isn't going to pop up in nature. Things like the existence and uniqueness theorems for ordinary differential equations are practically useless, because they only apply to things that physicists never really doubted in regards to having a solution. Classical EM predicts that the self energy of an electron is infinite. This problem still exists in quantum theories, but we get around it by doing some voodoo hand-wave magic.
Logic is a tool, and a vitally important one at that. But you don't get progress in math, physics, or any of the other sciences without people having good intuitions and hunches. Even philosophical considerations have affected the history and development of science, and those are ideas that ultimately are axiomatic and not fundamental consequences of logic.
I like your points but I dont see how logic can be likened to a man made tool. It doesn't matter whether the logical status was clarified. For example emotions leading people to "good intuitions and hunches" was all based on top of logic operations. This was still true 20k years ago before humans understood much about it. There is nothing below logic unless you believe in some form of mysticism.
Last edited by PC2; 2016-12-09 at 09:36 PM.
We are inherently subjective beings and anything like objective logic cannot, in truth, be concocted by us. There will always be a degree of subjectivity involved in anything we do, simply because it is us doing it. That said, the organisation of logic is usually preferable to the raw emotion of feelings.
There is nothing below logic that we can formalize. What logical operations were involved in determining that we should study shapes? What logical operations can be used to define the fundamental concepts of plane geometry? None, because it is impossible.
How do you define a point, or a segment in plane geometry? How do we define time and motion?
The reality is that everything we know is built up from undefined notions that we simply rely on people to intuitively understand. Because any attempt to pin down these concepts with logic leads to circularity and infinite regress. Everything we have is built up from these using logic and oftentimes more intuition.
There is absolutely something more fundamental than logic, namely the reality that we experience, or the primitive notions in mathematics. Logic is a process, and a process needs something to proceed from. And that thing, whatever it is, is more fundamental.
Last edited by PC2; 2016-12-09 at 10:00 PM.
Feeling gives insight to the desirable result an individual wants. Babies and cute children appeal to many people's emotions and most want the best for them. There is also self interest at stake as well. "Who cares what happens to people across the ocean, I don't want my new iPhone to increase in price!"
Logic tends to come after and hopefully molds the thought process to reasonable and positive results.
The wise wolf who's pride is her wisdom isn't so sharp as drunk.
Infinite regress doesn't occur because of complexity. It occurs with simplicity. The examples I listed above are a few of the most fundamental notions in math or physics. Almost everything in plane geometry is defined in terms of points. But how do you define a point? You can't define it in terms of the derived notions, as that would be circular. If you found something else that you can use to define a point, then how is that something else to be defined? It simply pushes the same problem back another step; the infinite regress.
As I said, logic is only a process. If you don't have a collection of propositions and basic relations among then, then what good does logic do you? What good is "P and Q implies R" if there isn't even a P, Q or R? The primitive notions of mathematics are more fundamental, because without them logic is a useless thing. There is nothing to apply logic on. With basic notions of time, position, etc, what can logic really do in science? Nothing.
Logic is like a flashlight. It needs batteries to work. Without the batteries, the flashlight is useless. Without the flashlight, you still have the batteries.
It doesn't prove that it's probabilistic, and it also doesn't prove that it isn't. Physicists are generally fine with this because how we choose to interpret it is usually independent of how it is actually used; i.e. 'shut up and calculate.'Not being able to predict quantum events doesn't prove it is probabilistic. It just means it's not currently understood.
It may mean that we don't understand. But it may also mean that true probability exists. We really don't know and have no way of knowing right now. The underlined sentence is not a logical conclusion of science, but a logical conclusion based on science and some sort of intuition about how the universe should be. One could equally well argue that determinism is a trick, an emergent property of underlying probability.
There shouldn't be politics.
I believe that falls under Logic and Pragmatism.
All right, gentleperchildren, let's review. The year is 2024 - that's two-zero-two-four, as in the 21st Century's perfect vision - and I am sorry to say the world has become a pussy-whipped, Brady Bunch version of itself, run by a bunch of still-masked clots ridden infertile senile sissies who want the Last Ukrainian to die so they can get on with the War on China, with some middle-eastern genocide on the side
Where's ethics & principles
If you are particularly bold, you could use a Shiny Ditto. Do keep in mind though, this will infuriate your opponents due to Ditto's beauty. Please do not use Shiny Ditto. You have been warned.
Yup.
The reality of real numbers is actually a really interesting thing to consider. At least with integers and terminating decimals (some fractions), we have models that we treat as integers, i.e. whatever it is that computers are actually using when they use integers. But arguably nobody has ever seen an irrational amount of something.