109

Since you have asked for a non-formal answer, I shall try to oblige by not using any numbers or equations. Fundamentally, your question is, how does it come about that individual events can be completely unpredictable but when you pile a lot of them together, either in a sequence or in a mass, the behaviour of the whole pile becomes, if not totally ...


109

Is the world in chaos now? Because one plus one is not equal to two, at least not all the time. Take one liter of water and one liter of sand. Add them together. What do you get? Wet sand, but certainly not two liters of it. Take one rabbit and add one rabbit. Add them together. You have a reasonable chance of ending up with quite a bit more than two ...


102

As any mathematician will tell you, 1 + 1 = 2 follows trivially from definitions, and is not a theorem. Your question makes no sense. It is as though you declared: I define 1 fluid zounce to be exactly 30 millilitres. But what if it turns out I'm wrong? It is your definition. It cannot be wrong because fluid zounces, prior to your definition, ...


59

Complex numbers are not, as you suggest, "...an integral part of physical reality". Neither, as you say, does the "quantum wave distribution function necessarily uses complex numbers". Not necessarily. Quantum mechanics can be mathematically formulated using the real numbers, the complex numbers, or the quaternions. See, e.g., https://arxiv.org/abs/1101.5690 ...


57

In math, we define stuff like numbers and operators, then we go on to prove other stuff from those premises. When you ask: "Is 1 + 1 = 0?", a mathematician will just ask back: "With what definition of +?" If you assume natural numbers and the common definition of +, then this statement is false. If you assume numbers modulo 2 and + meaning XOR, then this ...


42

If the probability of heads = p , then the probability of tails = 1-p . If it's a fair coin, then p = 1-p and the probability of either heads or tails is p = 1/2. Now suppose the number of coin tosses is N, and let's say that N is getting pretty large. The expected value of the random variable that is the number heads out of the N tosses is going to be ...


35

Ill formed question. Mathematics (specifically, logics) define what truth is. You are trying to test the validity of the tool with the tool itself. The answer would be a plain "yes". Otherwise (if you discuss mathematics as an issue of perception) you fall into Rusi's answer. Yes, you can have i apples, if you define the domain of i (i is not just a ...


33

Physical sciences rely upon thinking of hypotheses and testing them with experiments. The conclusions from physical sciences are always scrutinized because it is the way of the scientific method. In order for a scientific theory to become better, first a deficiency in the theory is discovered, followed by an altered hypotheses, followed by re-testing. ...


28

Is First Order Logic (FOL) the only fundamental logic? Short Answer No. It's just the most popular logic among mathematicians and philosophers for primarily historical and cultural reasons. Long Answer Since you wrote a long question, here's a long answer :-) Originally, Frege proposed a form of second order logic as a foundation for mathematics in his ...


28

After a bit of searching, I found some promising leads (and quite a few consistent descriptions) which suggest that Russell thought Gödel's results were of cardinal importance, but misunderstood their implications. In particular, he thought that Gödel's result essentially entailed that Peano Arithmetic was inconsistent rather than incomplete; but ...


28

Yes. Some will say that a proof is defined simply in purely technical and syntactical terms: a set of statements that conforms to a certain set of syntactical transformation rules. As such, you could even have proofs with no mathematicians at all. However, even if you add the 'convincing' part in there, I would say that when people produce proofs, they are ...


27

Consider the following analogy. What is a chicken? Are chickens real? There was a time (most places in Europe, anyway) when this would have seemed an even more stupid question than it does now. Everyone knew exactly what a chicken was. Even a rich noble would have only had to walk perhaps fifteen minutes and point to an example of a chicken. It was a ...


25

The convergence appears pretty quickly. This is your faulty assumption. It does apear pretty quickly. In most cases. But not at all every time. There are in some sense two layers of likelyhood: In layer one, every single event has the very same probability as its precedessors. In layer two, the sequence of events as a whole has a probability to occur. And ...


24

Why is 2 considered a prime number? This is really a question of terminology. The current notion of an integer that is unrepresentable by a product of other integers is given the name "prime number," and you're asking why the term "prime number" doesn't refer to some other set of numbers. Ultimately, it comes down to what is useful to mathematicians. ...


24

According to Peano's axioms zero is the number which is not the successor of a number. For each natural m, addition by n is defined by induction on n: m+0 := m, m+(n') := (m+n)' here the symbol ' denotes the successor. According to von Neumann's definition of natural numbers, the number zero is the empty set, and the number 1 is the set with single ...


23

This is not a mathematical argument, so no mathematical response is necessary. Using the standard axioms of set theory and the standard mathematical definition of "cardinality", it is an absolutely true statement that the cardinality of the even numbers is the same as the cardinality of the integers. One can argue about whether the notion of cardinality ...


23

The only thing you have to assume to be unconditionally true in Mathematics is some minimal logic (and yes, that's despite having axiomatic systems for logic; you still have to use some form of logic to actually define those axiomatic systems). But logic is assumed to be true in any science (because without it, you cannot draw any conclusions). But apart ...


23

Aristotle's solution was largely accepted until the end of 19th century when Cantor and Dedekind formalized the notion of continuum in terms of set theory. Under their interpretation time is in fact composed of indivisible nows, just like a line is composed of points, and any other magnitude is composed of indivisible elements as well. It does not mean that ...


22

The other (perfectly good) answers reason from how prime numbers are usually defined in Mathematics. I will approach your question in a different way -- seeing what your definition leads to. I think a prime number is a number that is divisible only by itself and one [exclusionary remarks omitted, for the sake of argument]. [A] prime number must be able to ...


22

In broader mathematics, the defining property of 0 is that it's the additive identity — that is, adding zero to another number doesn't change that number. This isn't inherent in the Peano axioms. The Peano axioms simply say that there is a natural number which isn't the successor of any other natural number, and that the symbol 0 represents that number. In ...


21

First of all, I'm surprised this was migrated from mathematics; it seems to me to be a much better fit there than here in philosophy. That being said: In short, his point is that it doesn’t make sense to talk about the set of even numbers and the set of all integers as if they were two separate sets. They are essentially the same set, only with different ...


21

I think it is a mistake to assume that there exists something like a context-independent notion of truth. Let me explain what I mean with the context dependence of truth. Consider the following simple question: Did Han shoot first? Now you can observe that in the real world, as far as we can tell, Han didn't exist at all. Obviously a person that doesn't ...


20

The short answer: Your premise is not correct. Quantum Mechanics is not necessarily complex-valued. Here is a primer from Physics.SE if you are solid on the math. An explanation that is light on math: Complex numbers represent a particular collection of symmetries that behave in a particular way. They happen to be closely related to Real numbers because ...


20

This is a semantic problem. There are multiple operating definitions of "proof," and the textbook you're reading from fails to distinguish between them. That's okay in a practical sense, but it's not okay if you're asking a question like this one. So let's split it up. There are lots of other potential operating definitions that are compatible with the one ...


19

Leitgeb distinguishes between statements, which are declarative sentences (he calls them 'descriptive sentences'), from propositions, which, unlike statements, are not linguistic objects. Propositions are the sort of objects that can have truth-values. E.g., [that snow is white] is a true proposition (Lecture 2-1). Once the distinction is made, the key idea ...


19

Yet we also know that the series will converge upon an equilibrium of heads:tails. I think this is your central problem. This is indeed the most probable result of a series of coin tosses, but probability doesn't apply to things that are already known to have happened. Imagine this game: A coin is tossed 100 times. Gamblers can bet on the total number ...


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