# Tag Info

### 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 ...

### 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 ...
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### 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 ...
• 1,460

### 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 ...
• 3,187
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 ...
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### 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 ...

### 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 ...
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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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### 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 ...
• 169

### 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 ...
• 1,772

### 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 ...

### What is the difference between the complex numbers i and -i?

Since i is defined as a solution to the equation x2 + 1 = 0, and −i also satisfies that definition, there is no algebraic fact about i which ceases to be true when all instances of i are replaced with ...
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### What is the difference between the complex numbers i and -i?

You say: So how can we "witness" a property distinguishing i and -i? The answers present here all essentially and convincingly acknowledge that taken in isolation, i and -i are ...
• 29.6k
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 ...
• 2,809
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....
• 35.7k

### 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 ...
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### What is the difference between the complex numbers i and -i?

== I'll give this a shot. You could be looking for the mathematical difference or the physical difference. I'll tell you about the physical difference. On the imaginary plane, things below the real ...

### What is the difference between the complex numbers i and -i?

One possibility is "Arg(i) is positive, and Arg(-i) isn't", assuming that you are comfortable with the distinction between positive and negative numbers. And if the Arg() function displeases ...
• 566

### What is the difference between the complex numbers i and -i?

If i and -i were identical, then you could substitute one for the other in any mathematical identity. For instance, we have the identity i × -1 = -i. If we substitute, we get i × -1 = i, which is ...
• 143

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

It is worth noting that in a strict mathematical sense, plenty of things that we think of as comprising a set, e.g. the members of a football team or the apples in a basket, are not actual sets. They ...
• 1,602
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 ...
• 28k

### 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 ...
• 28k

### 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 ...
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### 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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### 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⁻ (...
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### Thomson's lamp: a useless paradox?

You ask: Is there any depth to this famous paradox beyond such a conflation of kinds of infinite number? Yes. It plays a role in metaphilosophical discourse in speaking to the power and limits of ...
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### What is the difference between the complex numbers i and -i?

The complex numbers aren't simply a 2d vector space. They also come naturally with a basis: 1 & i. Thus they are examples of framed vector spaces. Hence i & -i are distinguishable by use of ...

### What is the difference between the complex numbers i and -i?

Taken just so far, the sequence of in does not reveal the fuller significance of its terms. That is, if we have thought our way to just i and -i, as such, we would not have a way to attribute to them ...
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