66
votes
Accepted
Is infinity a number?
Infinity is not a real number. All real numbers x have the property x + 1 > x. Infinity does not share this property.
Infinity is an element in the system of extended real numbers. However, this ...
- 1,576
42
votes
What is the difference between the complex numbers i and -i?
The key is that the field of complex numbers C has a nontrivial automorphism group over the field of real numbers R: the nontrivial automorphism is the complex conjugate. So whatever construction you ...
- 421
39
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, ...
- 640
36
votes
Is the surprising applicability of mathematics to the physical world a brute fact, or something crying out for a theistic explanation?
Reality existed.
Math was invented, partly to describe and predict reality, a useful tool.
Calculus specifically is an example...
Isaac Newton (1642–1727) is best known for having invented the
...
- 1,736
36
votes
Is there a paradox in the proof of Godel's incompleteness theorem?
Gödel was right. O'Connor 2005 meets every known objection: it is constructive and finite, indeed it runs on commodity hardware in reasonable time; it includes Rosser's trick, it is not relative to ...
- 1,650
36
votes
Is infinity a number?
It depends entirely on what you mean by "number." You might be surprised to learn that there is no standard definition of the word "number" in mathematics! Instead, there are many, ...
- 1,488
27
votes
Accepted
Does Math use the scientific method?
I'm going to go against common opinion here and say that, while mathematical truths may be considered distinct from scientific truths (and I'm not disagreeing with that), the process of discovering ...
- 3,769
25
votes
Is it possible to do physics without mathematics?
At about 1600 CE Galilei made his famous statement that the book of
nature is written in the language of mathematics. At least since
this time it was beyond doubt that progress in astronomy and ...
- 36.9k
24
votes
Accepted
Is the surprising applicability of mathematics to the physical world a brute fact, or something crying out for a theistic explanation?
The biggest issue seems to be that Craig implies that mathematics is entirely disconnected from the physical world. But maths emerged from our understanding of physical world. We saw that when you put ...
- 11.7k
24
votes
Is there a paradox in the proof of Godel's incompleteness theorem?
You do not understand the incompleteness theorem. It does not require "coding", and it does not depend on "actual infinity", and it does not "hide" any paradox. You ...
- 851
21
votes
Is infinity a number?
You have run into a common situation in philosophy, where you asked a question using a word that is weak to that question.
The existing answers are good, but miss an opportunity to get better at ...
- 388
21
votes
Is it possible to do physics without mathematics?
It depends on what you mean by 'do' physics. You can certainly get a grasp of key ideas of physics, such as relativity and electromagnetism, without resorting to mathematics, but real physics, of the ...
- 24.8k
19
votes
Accepted
If physics can be reduced to mathematics (and thus to logic), does this mean that (physical) causation is ultimately reducible to implication?
Physics, or indeed any other science, does not reduce to mathematics. Rather, physical relationships are expressible in the language of mathematics. If you wish to state Coulomb's Law, you can use a ...
- 29k
18
votes
Can LLMs have intention?
No, an LLM itself is a statistical model stored as a artificial neural network about linguistics in a training set. It has no activity, nor intentions. It has no perception of the world or time, no ...
- 5,705
18
votes
Accepted
What do all branches of Mathematics have in common to be considered "Mathematics", or parts of the same field?
Instead of asking what all branches of mathematics have in common, I'd look at that as: how can I distinguish a mathematical text (proof, argument, construction) from a non-mathematical one. I believe ...
- 2,570
17
votes
Thomson's lamp: a useless paradox?
The resolution to the paradox is that it violates the laws of physics. It would take an infinite amount of energy to move the lamp switch so fast. "Would the lamp be on?" is a question about ...
- 16.5k
17
votes
What is the difference between the complex numbers i and -i?
I would probably say that there is really no essential difference between the number i, considered in isolation, and the number -i, considered in isolation. If somebody hands us a system, and we ...
- 1,488
16
votes
Accepted
What is a philosophical interpretation of Bayes’s theorem when one of the probabilities is zero?
Bayes' Theoreom has P(E) != 0 as one of its pre-conditions. As with any other theorem, if any of the premises is not true you cannot validily deduce the conclusion.
So the math thing to say is not &...
- 522
16
votes
Is infinity a number?
The answer to your question is context-dependent and definition-dependent; it varies between different areas of mathematics. In ordinal arithmetic, for example, the first transfinite number, ω, can be ...
- 683
15
votes
Can Mathematics Fully Describe the Universe?
Clearly, no mathematics is ever going to describe redness and pain and love to any reasonable person's satisfaction.
To describe something is to say what you think this something is by using a subject-...
- 9,038
15
votes
Accepted
Can axioms be false?
It depends. An axiom is a starting point or foundation for reasoning. The term has been used in different ways historically. It used to mean a proposition that is self-evidently true, or at least so ...
- 29k
15
votes
What is the difference between the complex numbers i and -i?
If I understand Stewart Shapiro in his Identity, Indiscernibility, and ante rem Structuralism: The Tale of i and −i correctly, his, "there is no requirement that mathematical objects be ...
- 3,370
15
votes
What do all branches of Mathematics have in common to be considered "Mathematics", or parts of the same field?
Mathematical formalism gives an easy definition of the field. For example, Haskell Curry defined mathematics as the "science of formal systems". From this perspective, it's easy to test if a ...
14
votes
If physics can be reduced to mathematics (and thus to logic), does this mean that (physical) causation is ultimately reducible to implication?
Bumble's answer is spot on, but I thought I'd show you the fallacy of your reasoning a little more concisely.
You are engaging a fallacy called reification. From WP:
Reification (also known as ...
- 31.4k
13
votes
What is a natural number?
As far as I can tell, Peano's Axioms arose from properties of the counting (natural) numbers that have been known for thousands of years:
Zero is a natural number
Every natural number has a unique ...
- 1,286
13
votes
Thomson's lamp: a useless paradox?
==
The paradox arises from confusion between an open set and the closure of that set. It also makes the common error of confusing infinity with a large integer.
The total time that the switch is being ...
- 854
12
votes
Does zero exist?
You can make a case for zero being a simple idea, rather than the esoteric creature you take it to be. If I have two eggs and eat them both, I no longer have any eggs- that is a concept a child could ...
- 24.8k
12
votes
Does Math use the scientific method?
The relationship between mathematics and the scientific method is a complex and debated topic. Here's my perspective on it:
Pure mathematics, which deals with abstract concepts, axioms, and logical ...
- 121
12
votes
Accepted
Is there a difference between "set" and "collection"?
A "collection" is a general word to refer to "some things", without specifying a formally described structure such as a set, class, type, conglomerate, etc.
Mathematicians ...
- 3,152
11
votes
Does Math use the scientific method?
In my view, there is a false dichotomy muddying this issue. No, mathematics does not use the scientific method in the same sense as the natural sciences. The results of experiments do not count as ...
- 299
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