As a mathematician, it's quite interesting to read and discuss philosophical induction, because it does lack certainty and rational. A physicist will have you believe the sun will rise tomorrow not because it has risen for the past 10,000 years, but because of the models of physics which have such a great body of evidence to support, but these theories are still solely an inductive argument based upon observation, no matter how exacting they may become. Gravity, mass, and motion are the cause behind the sun rising and these causes must continue as long as nature is uniform. As you essentially stated, the whole of science tests on the condition that nature is uniform. In mathematics, we can squeeze the probability of induction into certainty. We deductively prove that one step of the induction leads to the next step of the induction, and then we show that we can take the first steps of the induction without problem. I think the certainty that mathematics provides is one of the reasons I am drawn to it.
Mathematics is certain up to an extent. "1 + 1 = 2" is true not just because of inductive observations about seeing one object and another object, but because mathematicians have formally defined what "1 + 1 = 2" means using various different axiomatic methods. These basic building blocks which follow logical principles allow for proofs of certainty. Of course, these proofs are entirely dependent upon the axioms being true. Godel's Incompleteness Theorem famously tells us that parts of mathematics are not provable inside the world of mathematics. They only can be true or false if we define them as such axiomatically. We cannot be certain that a conjecture will be proven true or false, but once a theorem is proven true or false, we can be certain of that conclusion, as long as the conclusion comes from the given axioms. In a sense, mathematicians are a kind of God of their universe. When it comes down to it, we define the exact rules which govern things and have designed those rules so that they cannot change. In this metaphor, a mathematical sun is guaranteed to rise tomorrow because part of the definition of a mathematical sun is that it rises tomorrow.
If you have ever been arrested and shown overwhelming evidence of your guilt, and you know none of it is true about you, you are being convicted with inductive reasoning and circumstantial evidence. The 12 jurors are totally hoodwinked all too often into buying the prosecution's story. There is no past and no future. There is only an eternal now. Ignore your ego's lamentations.
As a mathematician, it's quite interesting to read and discuss philosophical induction, because it does lack certainty and rational. A physicist will have you believe the sun will rise tomorrow not because it has risen for the past 10,000 years, but because of the models of physics which have such a great body of evidence to support, but these theories are still solely an inductive argument based upon observation, no matter how exacting they may become. Gravity, mass, and motion are the cause behind the sun rising and these causes must continue as long as nature is uniform. As you essentially stated, the whole of science tests on the condition that nature is uniform. In mathematics, we can squeeze the probability of induction into certainty. We deductively prove that one step of the induction leads to the next step of the induction, and then we show that we can take the first steps of the induction without problem. I think the certainty that mathematics provides is one of the reasons I am drawn to it.
Is mathematics certain? How do you know?
Mathematics is certain up to an extent. "1 + 1 = 2" is true not just because of inductive observations about seeing one object and another object, but because mathematicians have formally defined what "1 + 1 = 2" means using various different axiomatic methods. These basic building blocks which follow logical principles allow for proofs of certainty. Of course, these proofs are entirely dependent upon the axioms being true. Godel's Incompleteness Theorem famously tells us that parts of mathematics are not provable inside the world of mathematics. They only can be true or false if we define them as such axiomatically. We cannot be certain that a conjecture will be proven true or false, but once a theorem is proven true or false, we can be certain of that conclusion, as long as the conclusion comes from the given axioms. In a sense, mathematicians are a kind of God of their universe. When it comes down to it, we define the exact rules which govern things and have designed those rules so that they cannot change. In this metaphor, a mathematical sun is guaranteed to rise tomorrow because part of the definition of a mathematical sun is that it rises tomorrow.
If you have ever been arrested and shown overwhelming evidence of your guilt, and you know none of it is true about you, you are being convicted with inductive reasoning and circumstantial evidence. The 12 jurors are totally hoodwinked all too often into buying the prosecution's story. There is no past and no future. There is only an eternal now. Ignore your ego's lamentations.
What is this "tomorrow" you speak of, as it even exists? 🤔😊