110

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 ...


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 ...


101

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, ...


83

My personal point of view is that mathematicians invented the axioms and the rules of operation, the rest are discovered. Mathematicians invented the notations for writing down the concepts which are discovered within the universe of an axiom. The concept of numbers exists, but we invent the notation that the glyph '1' and the sound /wʌn/ refers to the ...


58

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 ...


39

The literature on these questions is immense, starting from Plato all the way to the modern mathematical logicians. Since your question is about the existence of numbers, you are concerned with the ontological status of numbers. So, with ontology in mind, you can distinguish the following schools of thought, according to the answer they give to your question....


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. ...


33

Mathematicians find the property A non-invertible number that cannot be written as the product of two non-invertible numbers to be more useful than the property A non-invertible number that cannot be written as the product of two non-invertible numbers and is additionally larger than some other non-invertible number That's why the former property has a ...


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 ...


28

If you want to take a more constructive point of view, you need to reinterpret things accordingly. For example, "not P" should be interpreted as the assertion "P implies a contradiction". Accordingly, Fermat's last theorem says: Given any solution to xn + yn = zn, n > 2, you can deduce a contradiction To prove this statement, you don't need to examine ...


27

This is only a partial answer: As a mathematician, I have been asked this sort of question from time to time. Like most other mathematicians, I tend to sort of evade the question, because it's tricky. Usually, the question is put in the form, "Are you a platonist?" The reference here is to Plato's eternal form that we are able to recognize, and that allows ...


27

Any effectively axiomatized formal system that extends a very basic theory of formal arithmetic called Robinson Arithmetic (Q) will contain an undecidable sentence. In full generality, you can state the syntactic version of the First Incompleteness Theorem as follows: (G1T) For any effectively axiomatized theory T that extends Q there exists a T-sentence ...


27

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 ...


26

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

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 ...


24

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 ...


24

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. ...


23

This depends very much on the area of philosophy. If you're interested in philosophy of quantum mechanics, for instance, you need at least undergraduate level training in physics (and the mathematics that entails). If you're doing ethics or political philosophy, then maybe the need for that sort of knowledge is lessened (although knowledge of some basic ...


23

Within the subject of rational choice theory, there has been an extensive axiomatic development of various rational decision theories, in which general principles of rational decision-making are put forth in a general context, and then detailed arguments are made to deduce further consequences from them. A major issue had been the extent to which the ...


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

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 ...


22

We don't know. There are some very valiant attempts to engage the question here, and many of them even explore concepts well worth exploring. But just because we live in such a complex, information-packed age doesn't mean we need to pretend we know things we don't. The oracle at Delphi said that Socrates was the wisest man in Athens simply because he ...


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 ...


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