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ALine

Question about proofs and logic

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Hey whats up,

question,

Is there some underlying linear understanding for how one may go about understanding mathematical proofs?

for example

Definitions -> Postulates -> Theorems -> Proofs -> etc.

Like is there a universal path of understanding for some logical statement?

The reason I ask is because when reading a little of "Journey into Mathematics" and the Elements it would continuously go through this process like one thing is built on top of another. That is cool and all but is there like an existing quantifiable formula for this process?

Thank you for your time

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17 minutes ago, ALine said:

Hey whats up,

question,

Is there some underlying linear understanding for how one may go about understanding mathematical proofs?

for example

Definitions -> Postulates -> Theorems -> Proofs -> etc.

Like is there a universal path of understanding for some logical statement?

The reason I ask is because when reading a little of "Journey into Mathematics" and the Elements it would continuously go through this process like one thing is built on top of another. That is cool and all but is there like an existing quantifiable formula for this process?

Thank you for your time

 

Surely this belong with your other thread on the subject?

 

Anyway here are a couple of thoughts on the subject, extracted from a couple of books on the subject.

Congratulations on being prepared to study books.
Sadly a quality in short supply these days.

Anyway first is from a famous texbook from lectures given at Cambridge University on Real Analysis.

Secondly is from a modern text on Geometry, which has changed a great deal since Euclid and become largely algebraic.
I post it because it is the bit about the axioms of vector spaces and shows a modern prsentation of what you ask in a vitally important subject, linear and affine maths.
This goes someway towards your desire for a flowchart.

It should be self evident which is which. So ask or discuss away after you have read them.

Lwood1.jpg.1bdfa3ac96368e1afe50a1174676571c.jpg

Lwood2.jpg.d01f061675046a209d60d795d3b1d953.jpg

Roe1.thumb.jpg.10e0053804f8a2d6429de4db2f9474b8.jpg

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So if I am reading this right, also look at the wiki, affine spaces are single point representations of linear transformations between points on a Euclidean space. Like if you have deltaX  = X2 - X1, in this affine space this would represent a single point?

Also, good read, however I am having a few problems fully absorbing the information of the first text.

Thanks for the info

Also if the above statement in which I have made is true then would that mean that because they are only transformations points in affine space would be real if the points in the Euclidean space are also real?

20 minutes ago, studiot said:

Surely this belong with your other thread on the subject?

Also Apologies for not keeping it to a single thread, will try to remember for next time.

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Consider a straight line.

A straight line through the origin is linear.

A straight line not through the origin is affine.

 

This is because of the constant

y = mx  is linear

but  y =mx +c is affine

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(Also the statement "A systematically purist style is out of the question" peeked my interest, why is this so? It gives the inclination that for the formal definitions of mathematics and logic , one must use a systematically purist understanding in order to concieve of the notions of each field, however once you actually look into each field a pragmatic approach is taken. The latter statement makes sense, because if you were to attempt to develop new systems outside of the previously defined axioms then there would inevitable chaos. Or am I incorrect in making the previous statement?) < please do not answer this question until I am less ignorant of the field.

 

apologies for jumping around

22 minutes ago, studiot said:

This is because of the constant

y = mx  is linear

but  y =mx +c is affine

so every straight line which is not passing through origin, that being (0,0), is affine?

Does that mean that every line which does not pass through the origin is affine?

Is that why in linear algebra all the lines come from the origin? 

So is normal algebra affine algebra? 

Are non-linear equations all just apart of affine algebra?

What else am I missing.

I have so many questions now.

Edited by ALine
put parenthases around the first statement

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1 hour ago, ALine said:

Is that why in linear algebra all the lines come from the origin? 

You must have really lucked out if that happened in your linear algebra class.

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1 hour ago, ALine said:

Hey whats up,

question,

Is there some underlying linear understanding for how one may go about understanding mathematical proofs?

for example

Definitions -> Postulates -> Theorems -> Proofs -> etc.

Like is there a universal path of understanding for some logical statement?

The reason I ask is because when reading a little of "Journey into Mathematics" and the Elements it would continuously go through this process like one thing is built on top of another. That is cool and all but is there like an existing quantifiable formula for this process?

Thank you for your time

The answer...is a bit complicated.

From the other thread, I'm guessing that your level is around a freshman or sophomore undergraduate with a new interest in math. At that level, and for a few more years, that is the approximate idea: learn the definitions, understand the postulates (there's less of a difference between those two than you might think), play around with them a bit, see if you can figure out patterns for yourself, figure out or read theorems, learn the proofs, etc. In time, you will be able to figure out some of the standard proofs for yourself. (One of my favorite/hated lines from Munkres, chapter 4: "Why do we call the Urysohn lemma a ‘deep’ theorem? Because its proof involves a really original idea, which the previous proofs did not. Perhaps we can explain what we mean this way: By and large, one would expect that if one went through this book and deleted all the proofs we have given up to now and then handed the book to a bright student who had not studied topology, that student ought to be able to go through the book and work out the proofs independently.") To a large extent, the "game" is "Here are our assumptions; what are our implications?" 

When you have gotten comfortable with this, usually around the time you graduate from a math or math-related major, things start to change a bit. By that time, you stop simply accepting definitions and start asking "Why was this definition chosen, rather than that one?" The "game" shifts; definitions become more fluid, although rigorous proofs stay, in some sense, the center.

You may or may not have heard of Terry Tao; he's a very famous mathematician. He explains what I said above in much more detail here:

https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/

For now, it is a good idea to focus on learning the definitions and thinking in the "undergraduate" way. Getting to the "graduate" way takes a lot of investment and time that you haven't had the chance to put in yet, and knowledge that you haven't had a chance to learn yet. But you are well on your way through the first transition, from the sounds of it. 

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3 minutes ago, uncool said:

I'm guessing that your level is around a freshman or sophomore undergraduate with a new interest in math

Immm actually a senior, I have been doing electrical engineering up until now and decided to transfer because I was not doing so hot and that I was good at math. Proofs hit me like a freight train in understanding.

 

4 minutes ago, uncool said:

For now, it is a good idea to focus on learning the definitions and thinking in the "undergraduate" way. Getting to the "graduate" way takes a lot of investment and time that you haven't had the chance to put in yet, and knowledge that you haven't had a chance to learn yet. But you are well on your way through the first transition, from the sounds of it. 

Thanks mans

I just get this feeling that mathematics is like, as stated before, like everything else. From art to science to engineering, it is just more intrinsic. The formation of some thing using simpler things. But I will keep working at it until I get to that graduate level of understanding :D.

like one giant formless puzzle....thing.

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Posted (edited)

New question: I am working on a paper for my intro to advanced mathematics course and was wondering if coming up with relationships between sets is similar to coming with programs for a computer. The feelings when doing both feel the same, however I am unsure in respect to mathematics as a whole whether or not this is the "intention." Intention in referance to if I should be think about relationship deriving like I am writing a program.

Edited by ALine
Changed of to for, changed it to I

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Posted (edited)
2 hours ago, ALine said:

New question: I am working on a paper for my intro to advanced mathematics course and was wondering if coming up with relationships between sets is similar to coming with programs for a computer. The feelings when doing both feel the same, however I am unsure in respect to mathematics as a whole whether or not this is the "intention." Intention in referance to if I should be think about relationship deriving like I am writing a program.

Two subjects you might be interested in:

* Boolean algebra. That's the "algebra of logic and ,set theory" if you like. It makes some of the connections you're thinking of. A logical OR is plus, a logical AND is multiplication. The union of two sets is their sum, the intersection is the product. Products distribute over sums as they do with numbers, and so forth. Boolean algebra unifies a lot of different things that are really the same thing as you're sensing. 

https://en.wikipedia.org/wiki/Boolean_algebra

* The Curry-Howard correspondence and related notions. That says that computer programs and mathematical proofs are the same thing. This has repercussions in computer science and also philosophy.

https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspondence

 

Edited by wtf

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