All Questions
Tagged with philosophy terminology
25
questions
3
votes
2
answers
125
views
Creating larger structures from smaller ones without an explicit construction
I'm asking this question as a replacement for my previous one, which I admit isn't clear, and which I am voting to close. Hopefully I'll be clearer now.
Admittedly, I'm not sure if this question ...
- 585
1
vote
0
answers
178
views
When mathematicians say "true" do they mean "true in all models"?
According to the comments to this question,
Truth is ordinarily defined by reference to models.
If so, even axioms and theorems are not true without reference to a model.
However, when ...
- 103
-1
votes
1
answer
168
views
What are the differences between equality, equations and identities? [duplicate]
What are the differences between the followings:
Identity
$$
\sin^2(\alpha) + \cos^2(\alpha) = 1
$$
Equation
$$ 4x = 16 $$
Equality - $x,y$ are mathematical objects.
$$ x = y $$
All of the three ...
- 375
1
vote
0
answers
56
views
What makes one proof different from another one? [duplicate]
There are around 370 different ways to prove the Pythagorean Theorem, but what does that exactly mean? For instance, if your proof states that $x^2+y^2=z^2$, I could construct a different one by ...
- 33
0
votes
0
answers
66
views
Using set-builder notation in an unusual way for more concise expressions in a definition. Good style?
What do you think about the following usage of the set-builder notation $\{ x \mid P(x) \}$? Let $Y = \{ p_1, \ldots, p_n \}$ be $n$ objects and $A,B$ subsets of $Y$. Define a function from $\{1,\...
- 18.2k
-3
votes
1
answer
307
views
Why are inference rules necessary?
It has been stated, both on this site and elsewhere, that without inference rules, a theory can only prove its own axioms.
I don't understand how, though, and will try to show derivation without ...
- 1,129
1
vote
0
answers
64
views
Terminology for placeholder for elements of a set
Let $\mathbb{X}=\{1,2,3,4,5\}$ be a set and $x$ be an element of the set. Then 1 is also an element of the set. We say that $x$ is a "variable". I am searching for a word that conveys the ...
- 1,131
0
votes
1
answer
119
views
Analytic works versus synthetic works in mathematical research
With the purpose to clarify my ideas about terminology I would like to ask in Mathematics Stack Exchange what is the difference of two verbal expressions: when a professional mathematician understands ...
1
vote
0
answers
63
views
Is the length of a line a property of that line, or is it its own mathematical object?
I'm trying to understand the nature of mathematical objects.
As far as I understand it, mathematics studies these objects. Geometric shapes are one kind of such object, including 1D shapes, namely ...
- 11
1
vote
1
answer
67
views
Are there operations that can't be defined using a rule, and if they exist what is their significance?
Wikipedia:Operations(mathematics)
In mathematics, an operation is a function which takes zero or more input values (called operands) to a well-defined output value
What I took away from this fact ...
- 99
1
vote
1
answer
328
views
What is the term for a value that is directly dependent upon other values?
Say you have variables A and B, both of which express a certain quantity. Now let's say you have X, which equals the sum of A and B. For example, A = 10, B = 5, and X = 15 because 10 + 5 = 15. The ...
- 2,100
1
vote
0
answers
57
views
What is this process/action called in English?
it is a fairly generate question regarding a terminology.
People without science or engineering discipline makes an unfounded claim X, but people with such discipline start with proven facts A, B, C, ...
- 111
32
votes
7
answers
5k
views
Why do we call complex numbers “numbers” but we don’t consider 2-vectors numbers?
We refer to complex numbers as numbers. However we refer to vectors as arrays of numbers. There doesn’t seem to be anything that makes one more numeric than the other. Is this just a quirk of history ...
- 4,655
3
votes
3
answers
134
views
How do we say that two logically equivalent definitions are not semantically equivalent?
This is a question of terminology (and also relates to philosophy).
Very often we have a certain term, such as "differentiable function", and then we have a definition:
(1) def. A differentiable ...
- 13.4k
-5
votes
2
answers
215
views
In "10 grams of salt", is the unit "grams"?
The gram is a unit of mass, so "10 grams" has "grams" as the unit. "10 pounds" uses a different unit.
So what is the "salt" in "10 grams of salt", if not a unit? In other words, what is the ...
- 149