[Random self coined terminology] A set exists in concept space if there exists a map such that all its elements can be mapped to eulidean or noneuclidean space while preserving all the properties of the set
More mathematically speaking, given any concept, if I specify enough number of its properties (each property can be some nth ordered logic, and it does not need to be finite) then I will be able to localise it as a point or higher dimensional construct in some geometric space
THIS, is what I mean something that looks like space to me
So I guess, having such association might be actually suggesting I have a belief that every concept can be uniquely specified as a formula of its properties
Well, let's start with the context of abstract algebra
An identity map is defined to be a mathematical object that maps a class such that its image is the same as its domain
Or in symbols: $\text{id} : S \mapsto S, \forall x \in S, \text{id}(x)=x$
So, if we have the countably infinite parameter space (it has to be countable unless we use an infinite alphabet) where each parameter is a proposition like this, I can uniquely locate the object $\text{id}$ in this space as if this is some manifold with chart and I knew the coordinates of a point precisely
this is because anything that look like space is somehow more tangible to me in that I can manipulate them mentally
whether all this tendency for me to think of space will make me more receptive to category theory is currently an open question, especially, my own abstract algebra background is haphazard as many pointed out
Well, the simplest derivation of the inequality I know is to square both sides and move stuff around to get $(x_1^2+x_2^2)(y_1^2+y_2^2)-(x_1 y_1+x_2 y_2)^2\geq 0$
@Secret I think what you're talking about only covers concepts that you can think of. In that case, indeed, I agree with you that one can consider the 'space' of one's mental concepts. And as you notice, you're a bit stuck with what you can conceive of using whatever internal language you have in your mind.
@user21820 Indeed. Take art as an example, you can never specify all its properties, and good luck finding anything that can describe the life of one specific individual and reality itself
for a more closer to home example which I don't think is like space (or I am unable to map it): all the survival skills in a society
There is no clear reason other than a long history that explain why they took the form they are in the present
@Secret I see. I thought your claim was more ambitious, but I can see that you're not actually claiming much. I was thinking of various things that lie outside human perception and imagination.
@user21820 Theoretically, that framework might be able to account for those, but without a word to even describe it you cannot really illustrate such examples. After all, we humans are ultimately bounded by our hardwiring of the brain (and that is assuming consciousness is 100% physical hence the only reality we have is the one we are currently experiencing, the physical reality)
The exceptional cases correspond to either $\alpha=x_2/y_2=x_1/y_1=\infty$ (so that $y_1=y_2=0$ or to one of the ratios being formally indeterminate (0/0)
@Secret Actually, I don't buy the assumption that we are 100% physical, but even if we are, it doesn't change the fact that there are things beyond our description.
@user21820 This is one reason I hope within our lifetimes we will met extraterrestial civillisations, so we can learn all those alien worldviews that we can never comprehnd
The universe is big enough that alien life strikes me as both inevitable and irrelevant.
That is: The universe is so big that it's hard to believe that we're really the only sapient being in existence, but also so big that we are unlikely to ever come into causal contact with other civilizations.
And, regarding about your 100% physical comment, well, if consciousness really does have a nonphysical component, reality will become quite interesting, and sometimes scary.
Suddenly, you need to consider about someone you don't know that live in the opposite end of the earth on how it affect your life and other things
More generally, as I have found when talking to a neuroscientist x philosopher, in such a scenario, subjective realities and beliefs suddenly become something as physical as electromagnetic fields and you need to extend the notion of social interaction to such things and ho…
Compare that with the situation of explorers sailing across the globe and finding indigenous settlements: It was not inevitable that a given area would have been discovered and settled by then, and their existence (or lack thereof) was decidedly relevant to how those areas were exploited by their discoverers.
The difference being that the distance scales involved for sailing the Ocean versus travelling the stars are so vastly vastly different.
I guess you could say that we are not even equipped to discover life in different formats given how our definition of life is lacking given the information of our own biochemistry!
@LegionMammal978 the set of rationals to naturals can be put in bijection
The Earth feels big to us, but it's small enough that the possibility of 'alien' civilizations (from a Western perspective) could be both surprising and very relevant.
Could someone verify my proofs?
Proposition: the set of all finite subsets of $\mathbb{N}$ is countable
Proof 1:
Define a set $ X=\{A\subseteq\mathbb{N}\mid \text{$A$ is finite} \}$.
We can have a function $g_{n}: \mathbb{N} \rightarrow A_{n} $ for each subset such that that function is surje...
@user21820 The elements of life are everywhere, the time sufficient for evolution is there, our imagined self-importance is tricking us into believing that we are special.
@Secret My current guess is that our consciousness is intertwined with our physical self, but extends beyond the material world, and that we are permitted to interact with the physical world through our physical bodies, and that our brain is the primary interface connecting our bodies and our self. But it's just a guess and I won't attempt to justify it.
Sure. My point is again that, even if one is optimistic about there being life in the universe with which we could have meaningful interactions with, the probability of that interaction ever occuring is...not good.
I recently questioned all my beliefs (especially the faith) and dropped the majority of them, so these kind of conversations make me uncomfortable.. I don't really know why, I used to pretend knowing the answer to all of the problems in the frontiers of our thought, now, I admit my ignorance humbly, it's feels both so humiliating and thrilling.
Some numbers are useful. The distance traveled by the Victoria (the ship Magellan captained) was over 42,000 miles. The distance traveled by Apollo 1 to/from the moon was about 830,000 miles. The distance traveled by the Pioneer spacecraft so far is beyond 8 billion miles.
I agree, but i'm not sure that applies to some of the writers esp. some of the Hard Science Fiction authors, some are even qualified Astrophyscists and Mathematicians!
I’m going to ask a question that I probably should ask my professors, but I’m on summer break and so are they, and I want to know the answer to the question before the summer break ends.
Where can a person competent in both mathematics and programming find work that combines these skills?
In general, I’m not interested in analysis, stochastics, or related sciences, such as economics. Examples of workplaces within said fields would be much appreciated. If this question is too large, then is there any place that I can turn to besides my professors? My study councillor redirects such questio…
Okay. Well, I'll probably end up with about one years worth of university studies and about five years of algebra and related mathematics. Is that going to improve my stance against people studying pure programming? I was initially interested in analysis, stochastics, and computer science, but once I started doing my bachelor thesis in PDE I felt that I was displeased with the actual work of proving such theorems, and the applications within stochastics and computer science.
I want to have a plan B, and that was my plan B for analysis, but now I've become more interested in algebra. Thus, I want to know more in greater detail what my plan B will be when I specialise in algebra; I need to choose courses to be competent in plan B. Especially now as I'm going to study abroad at University of Melbourne I need to know it farther in advance.
@Semiclassical Yea I don't think it's meant to be speculative fiction rather than just a comic. But I recall there's plenty of scale-related panels like:
Okay. This is generally the understanding that I've been given; that programming involves lots of advanced data structures, algorithms, and mathematical concepts. I guess that I could email my professors, and look at the handbook for programming courses.
"Coding theory" seems to be my keyword when looking at what beloved Wikipedia has to say about it. :)
> The square roots of the prime numbers are linearly independent over $\mathbb Q$. (Proof: this is immediate given the ability to extend the function "number of powers of $p$ dividing $x$" from the rational numbers to algebraic numbers. $\sqrt{p}$ is "divisible by $p^{1/2}$" while any finite linear combination of square roots of other primes is divisible by an integer power of $p$, i.e., is contained in an extension of $\mathbb Q$ unramified at $p$).
I think searching Math SE will lead you to some of Asaf's answers about how taking away the axiom of choice prevents you from being able to prove certain things, and I believe it includes the existence of an uncountable Q-independent set of reals.
Of course, you don't need the full axiom of choice. You just need a well-ordering of the reals.
Hi there ,I am interested in competitive programming,and it involves a lot of mathematics although I am good at maths yet sometimes questions are a bit high than I am capable of doing.Any general advice to me .Thanks
@user21820 One place where such "chores" are handled nicely is at the C.R.U.D.E. chatroom. Anyway, there is a lot happening here; I could probably spend all my free time in various chat rooms, even on non.mse. chats!
@EmilioPisanty @ACuriousMind ...actually, I think I had all my physics thinking backwards. In general, I have this fixated conception that you can not only use experiments to tell you what model to construct, but you also can start with you model and figuring out what physics it corresponds to by performing known checks, providing the maths obey some known properties that the physical things are supposed to obey
Or in short, I thought the model contains all the information that can recover the physics of what each physical based mathematical object (e.g. self adjoint operators) will mean
@user21820 I think we've been doing great, myself. I left a comment at the CRUDE, just noting that it seems only two users that maintain it, rather conservatively.
@Secret I used to be a very regular user in this chat. So you can consider me an "alum". I started exploring other sites at a point when the main chat (this chat), mostly derailed from the basis of math; It is absolutely good to have a mix of math and socializing. But when chat users start attacking each other, etc., (as happened when I left), in can get ugly.
I see, in that case, based on what the chat is like in the past 5 months, I think you will quite enjoy the return. Maths is still the focus of this chat, but ocassionally, other socialising topics and other sciences such as chemistry and physics are common, especially one of the current regulars is a physicist, I am a chemistry PhD, some people sometimes ask chemsitry here and we also have some philosophers
Maths topic itself is not as focused on topology, and more balanced between a range of things
Currently, there is quite a lot of number theory as couple of users found interesting number theoric stuff and posted some MSE about them, they can be quite algebraic
Ted and tobias are stil around and tend to appear during the peak hours, balarka and topology people also appear regularly. The number theory people can appear throughout the day in more or less constant manner
Could someone verify my proofs?
Proposition: the set of all finite subsets of $\mathbb{N}$ is countable
Proof 1:
Define a set $ X=\{A\subseteq\mathbb{N}\mid \text{$A$ is finite} \}$.
We can have a function $g_{n}: \mathbb{N} \rightarrow A_{n} $ for each subset such that that function is surje...
