How to get LNC as a theorem using Frege's Prop Calculus?
So Im using axioms from,Frege propositional calculus and is there any way to derive Law of non contradiction as theorem from them.The axiomsA → (B → A) | THEN-1(A → (B → C)) → ((A → B) → (A → C)) |...
View ArticleWhy is it important to prove that some particular set is a vector space as...
In Axler's Linear Algebra Done Right Example 1.24, we are asked to prove that the set of all functions from some set S to the set of real (or complex) numbers is a vector space.I proved this by using...
View ArticleWhat are fun mathematical facts for non-mathematicians? [duplicate]
I like to spend my life with mathematics. I think it is the best thing I can do in my life. However, I have great difficulty explaining what I am doing to non-mathematicians, even educated ones. For...
View ArticleWhat does it mean to solve an equation?
This question might be more philosophical than mathematical.In school we are taught how to solve equations such as $x^2 - 1 = 0$ or $\sin(x) - 1= 0$. Solutions to these equations are quite simple. For...
View ArticleIs complex analysis more "real" than real analysis?
In physics, in the past, complex numbers were used only to remember or simplify formulas and computations. But after the birth of quantum physics, they found that a thing as real as "matter" itself had...
View ArticleFormally how do we view finite sets
This might be silly, but I have been thinking about how we would work with finite sets very formally.So, $\{1,2,3,\cdots,n\} = \{k \in \mathbb{Z}^+ \mid k \leq n\}$ gives a representee for which any...
View ArticleBenefits/uses of non-base 10 number systems?
For reference, I'm studying math and anthropology at university, and I've been dying to find some overlap of math theory and ethnomathematics (math uses/tools/systems/etc in other cultures). I'm...
View Articlemultiplicative number base?
Imagine a number base where instead of writing the number of products for each power of your base (e.g. in decimal, $152 = 1 \times 10^2 + 5 \times 10^1 + 2 \times 10^0$), you had one where you write...
View ArticleArithmetization of Turing machines
Refer to Turing's 1936 paper, page 248, last paragraph. I present the paragraph in verbatim below :The expression "there is a general process for determining..." has been used throughout this section...
View Articlewhy learning math in elementary school was harder for me rather than upper...
When I was an elementary student, I'd suffered from understanding basic things like multiplication table and other simple things and I had to memorized them. Last hours I was searching for genesis of...
View ArticleWhen 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 mathematicians say...
View ArticleWhy is negative minus negative not negative? Why is negative times positive...
What I understand is that the only difference between plus and minus is direction.I've never understood this: That +1 ++1 is +2, why is -1 - -1 not -2? (In the first, we are moving from right to the...
View ArticleWhat did Richard Dedekind mean exactly by his statement about generality?
But—and in this mathematics is distinguished from other sciences—theseextensions of definitions no longer allow scope for arbitrariness; onthe contrary, they follow with compelling necessity from the...
View ArticleWhat exactly makes the ordinals an indefinitely extensible concept?
I understand the principles of generation that cantor used to create the ordinals but I cannot see what exactly is the property that makes the ordinals an indefinitely open plurality and not the...
View ArticleZero-dimensional space with multiple objects
I am unsure if this belongs to math or philosophy.Let's say there's 0-dimensional space, however multiple objects exist within in, occupying the same "spot". If multiple objects exist, is the space...
View ArticleA philosophical question on the nature of mathematics [closed]
I had a seemingly simply question today, that goes as following.What do we need for a mathematics to exist in a universe, or a system, more broadly speaking?Is it a matter of having the ability to...
View ArticleDo we ever reason about a non-associative algebra without embedding it in an...
This question most certainly contains some errors in phrasing. It is on the subject of the philosophy of mathematics, and it is hard to stay precise when reaching towards the fundamentals of math.The...
View ArticleConceptual Question regarding Shannon Entropy and bits
It is said that the number of "information bits" contained in a certain piece of information can be roughly translated as the number of yes/no-questions that would have to be answered in order to...
View ArticleIs there a term for the idea that mathematical objects are defined by their...
In a recent Veritasium video discussing Euclid's Elements, Alex Kontorovich comments that Euclid's definitions of primitive objects (e.g. "A point is that which has no part.") are absurd and lead to an...
View ArticleHow irrational quantities physically exist in nature?
We know that an irrational no has well defined decimal values upto infinite decimal places. These irrational quantities exist in nature in some kind of measurements. For an example, circumference of a...
View ArticleFinitists reject the Axiom of Infinity - are there groups who reject the others?
I've seen rejections of the Axiom of Infinity. This is called finitism. Some ultrafinitists even add the negation of the Axiom of Infinity. Definitely doable.I've seen rejections of the Axiom of...
View ArticleWhy isn't finitism nonsense?
This is a by product of this recent question, where the concept of ultrafinitism came up. I was under the impression that finitism was just "some ancient philosophical movement" in mathematics, only...
View ArticleWhat is the logical system of Tractatus Logico-Philosophicus?
Tractatus Logico-Philosophicus states simply that6 The general form of the truth function is: $[\bar p, \bar\xi, N(\bar \xi)]$. This is the general form of the sentence.Wikipedia and other sources...
View Articledoesn't the independency phenomenon make a case for non-classical logic?
alright, this question is philosophical and somewhat fuzzy. i also admit to knowing little about logic. all in all, this question can possibly be easily resolved by either pointing to (perhaps even...
View ArticleFor every object, is there a unique notion of isomorphism?
Do you think that, according to most mathematicians, the following claim holds?(Claim) For every object, there is a unique notion of isomorphism.Perhaps one might think that for some sets, such as...
View ArticleA concept larger than any other value but not infinite.
Sorry for the philosophy, this might be better suited on that stack exchange, I do not know. Is there any concept which is larger than any given number but is not infinite and is comparable. I.e. $N...
View ArticleWhy pullback of ideal sheaf should be the conormal sheaf?
I'm sorry that this isn't really a math question, but this gap between my intuition and the truth bothers me. For closed subvariety (for simplicity) with ideal sheaf $\mathcal{I}$, the pullback...
View ArticleAre there any problems about the difference between set theoretic definitions...
I am a novice about this question, so if there is a misunderstanding then I apologize for it.As for Peano axioms, if I choose Zermelo natural numbers, and you choose von Neumann ones, then this doesn't...
View ArticleCan the set of computable numbers be used as a theoretical basis for calculus?
I recall from my Real Analysis course that the rational numbers $\mathbb{Q}$ are not suitable for doing calculus, and I believe the reason was that $\mathbb{Q}$ does not possess the least-upper-bound...
View ArticleMathematical Induction: Strong vs Weak Form
I have a rather naive question: The usual mathematical induction works by the same scheme: Let $n_0 \in \mathbb{N}$ a pos integer and $A(k), n_0 \le k \in \mathbb{N}$ family of statements. Then the...
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