# Tag Info

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 ...
• 1,559

### 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, ...
• 1,300

### 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 ...

### 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 ...
• 15.7k

### 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 ...
• 623

### 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 ...
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 ...
• 2,771

### Is math (only) a language?

I would like to bring this discussion down to earth. I am a retired professor who has taught fluid mechanics to engineering students. Many important fluid mechanics concepts (like vorticity) are next ...
• 237

### 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 ...
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### Is it fair to say truth is used more in logic than in math? If so, what are the reasons for doing so?

First, I think you're overestimating the force of 'truth' in logic. If we look at simple syllogisms like (as you suggested): P1: All men are mortal P2: Socrates is a man C: Therefore, Socrates is ...
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### Is infinity a number?

Many mathematicians (and perhaps authors of some of the other answers to this question) believe, following Cantor, that there are just two types of infinite number in mathematics: ordinal and cardinal....
• 1,949

### Is math (only) a language?

Looking at the the artifacts that mathematicians produce - basically, theorems and proofs and algorithms - mathematics is a demonstrative, non-empirical science. You could call that a language; the ...
• 1,514

### Is math (only) a language?

The word math, like nearly any other word in any natural language, can be (and is) used in many ways by many speakers in many contexts. For this particular speaker (me, a mathematician), if I am ...

### Is infinity a number?

Despite some historical doubts and temporary issues, infinity is studied rigorously in mathematics. That doesn't mean every single mathematician (or even aspiring student to give them some credit) is ...
• 3,060

### Is infinity a number?

A common reason we differentiate between infinity and real numbers in (introductory) calculus is because we define limits differently between them. We say lim x->0 f(x) = L if and only if, for all ...
• 333
Accepted

### Is it even possible to define "set" non-circularly?

If you use first-order logic, you formalize a language for expressing properties of things, and you have rules for making deductions from those propositions. Let's call a proposition a formula. Let's ...
• 2,771

### Thomson's lamp: a useless paradox?

It's similar to the Achilles-and-tortoise paradox: The situation is described as an infinite series of points in time, which converge on a given time t-limit; since the description does not define ...
• 159
Accepted

### Is math (only) a language?

The approach to consider math a language seems interesting, even if it does not capture all of the art of mathematics. Galilei’s statement considers math the language which encodes the laws of nature....
• 35.4k

### Is math (only) a language?

Most definitions of mathematics define it as a science or field of study. These definitions indicate that math is more than a language. Mathematical notation is a structured communication designed as ...
• 1,772
Accepted

### Can Internal Set Theory provide a complete system of arithmetic?

The basic premise of your question ("there are finitely many standard numbers") is incorrect. What is true in IST is a statement that does have a finitistic flavor, but it is a bit more ...
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### Is infinity a number?

Your title question is: Is infinity a number? In the body you say that you found a strong resistance from some folks on math fora against using infinity in math. Let me suggest that the title question ...
• 4,197

### Is infinity a number?

Mathematics is a game of definitions. In most fields of modern mathematics, you start with some things that you define to be true, which you call axioms, and ask what conclusions you can draw by ...
• 131

### Mathematical Realism and 0=1

I understand what you're getting at with the 0 = 1 thing. You're saying that we can represent negation of a proposition S as S → ⊥, where ⊥ is set to 0 = 1. And the empty set is the set with no ...
• 17.2k

### Justification for applied mathematics

You got it backwards. Humans came up with PA because they observed things in reality that could be represented abstractly in logical form. All you need is to believe that there is some model of PA⁻ (...
• 784

### Can every idea, including mathematical ideas, be reduced to a series of simpler ideas, without information loss?

There's a quote in Pascal's Pensees (#20-21) which seems closely related: (#20) Order.—Why should I undertake to divide my virtues into four rather than into six? Why should I rather establish virtue ...
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### Is it fair to say truth is used more in logic than in math? If so, what are the reasons for doing so?

I think you overestimate the difference. The branch of logic in which truth plays an important role is model theory. We may say of a sentence that it is 'true under an interpretation' or 'false under ...
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Accepted

### Some questions about the material conditional and entailment in intuitionist math

There are different ways of understanding intuitionism. For Brouwer, it is a theory about mathematical reasoning that is sharply at odds with the logicism of Frege and Russell. Mathematics is not ...
• 27.8k

### Is there a difference between "set" and "collection"?

To avoid the Russell antinomy modern set theory discriminates between “sets” and “proper classes”. The generic term is “class”. A class C is either a “proper class”, i.e. there is no class which ...
• 35.4k