112
votes
Why don't fair coin tosses "add up"? Or... is "gambler's fallacy" really valid?
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 ...
- 22.9k
109
votes
Accepted
What would happen if suddenly, 1+1=2 is disproved?
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 ...
- 1,214
103
votes
What would happen if suddenly, 1+1=2 is disproved?
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 ...
- 1,145
56
votes
Accepted
Falsification in Math vs Science
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 ...
50
votes
If math is so deductive, why is it so hard to discover new math?
If the rules of chess are so simple, why is it so hard to beat a grandmaster?
The answer is the 'combinatorial explosion'. You have a small and well-defined set of moves you can make at each step. Let'...
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43
votes
Why don't fair coin tosses "add up"? Or... is "gambler's fallacy" really valid?
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 ...
43
votes
Is mathematics politically and culturally neutral?
TL/DR:
Yes and no. Who writes papers and what they choose to research are products of culture, and in the US, our culture has a long history of racism. The actual theorems in the papers themselves are ...
- 2,818
41
votes
Isn't the notion that everything will occur in an infinite timeline an example of the gambler's fallacy?
It looks like you've hit upon the concept of almost surely in probability theory. Something occurs "almost surely" if it happens with probability 1, but there still exist situations where that thing ...
- 1,177
37
votes
Accepted
Why would infinite monkeys not produce the works of Shakespeare?
Yes, the monkeys will do it. No, they don't have to.
It's mathematically true that after removing all logistical constraints - which is what we mean when we say there are infinitely many monkeys, ...
- 620
35
votes
Is mathematics truth? As in the sense of that which is manifest or possible in reality?
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 ...
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29
votes
If there were only one single mathematician in the world, would they be able to produce a mathematical proof?
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 ...
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29
votes
Why can't numbers be 'used up'?
Does a song get 'used up' when we sing it? Does a story get 'used up' when we read it? Does a path get 'used up' when we walk it?
Forgive the computer science analogy here, but all of these things — ...
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28
votes
Why don't fair coin tosses "add up"? Or... is "gambler's fallacy" really valid?
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 ...
- 13.5k
28
votes
Accepted
Why do universities not teach constructive mathematics to CS undergraduates?
Let me offer a few thoughts, specific to mathematical pedagogy in computer science (in particular for the states):
(a): a typical BS computer science program barely has time to touch on computational ...
- 2,099
26
votes
Accepted
Is mathematics politically and culturally neutral?
From its inception, mathematics was intended to be independent of cultural contexts in the real world. For example, you could use the same rules of algebra and the same numerals to count your family ...
- 7,772
25
votes
How can we overcome the challenge of the anti statistical philosopher?
We will suppose for the sake of argument that there do exist an infinite number of parallel universes. The question then becomes "Which universe are we in?". We observed these events, so we ...
- 1,273
24
votes
Why is 2 considered a prime number?
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," ...
- 1,131
23
votes
Accepted
How is 0 defined?
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 ' ...
- 24.3k
22
votes
Why is the complex number an integral part of physical reality?
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:...
- 756
22
votes
Accepted
What is the idea behind "p or not p" being a tautology?
"P or not P" is a tautology of classical logic, but not of all logics. It is not a tautology of intuitionistic logic, for example. So, one approach would be to say that classical logic does ...
- 22.9k
21
votes
Accepted
Why is 2 considered a prime number?
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 ...
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21
votes
How is 0 defined?
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 ...
- 563
21
votes
What would happen if suddenly, 1+1=2 is disproved?
most fundamental equation
Your assumption is flawed. 1 + 1 = 2 is not an axiom of mathematics, but (as Sputnik points out) a consequence of the Peano axioms applied to base 10 representations of ...
- 319
21
votes
Is mathematics truth? As in the sense of that which is manifest or possible in reality?
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 ...
- 1,511
20
votes
Why don't fair coin tosses "add up"? Or... is "gambler's fallacy" really valid?
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 ...
- 309
20
votes
Accepted
What was Cantor's philosophical reason for accepting the infinite but rejecting the infinitesimal?
Here is Cantor in his own words (from his influential 1887 letter to Weierstrass):
"I begin from the supposition of a linear magnitude ζ which is so small that its product by n , ζ · n, for every ...
- 42.5k
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