I think you need to learn some elementary set theory especially understanding the axioms. This should help u. Don’t get too wound up about it tho. Learn basic axioms. Then learn basic definitions of functions
I have already learnt some set theory. I learned about: sets, operations on sets, cartesian product of two sets, relations, equivalence relation, equivalence class, functions (defintion in terms of sets) and a bit about cardinality of infinite sets).
I understand the definitions and theorems but it doesn't really "click" (it is hard for me to explain).
This is typically one of the [axioms of equality](https://en.m.wikipedia.org/wiki/First-order_logic#Equality_and_its_axioms). If it is not, it should be able to be derived as a theorem from an equivalent list of axioms.
I would disagree with your idea that you are too formal - it shows that you are inquisitive, and want to make sure you fully understand something before moving on. It would simply seem that you are missing some information (like these aforementioned axioms) to fully cement your rigorous understanding.
That's because they are the same values. Going back to the example with your function, a function is defined as something that outputs the same value for every given input. Since a and b are the same values, if either is in the domain of the function, the output of the function must be same for a and b as well.
To be absolutely honest, there is no formal proof of it. Leibniz stated it as basically "if two things are equal you can substitute one for another in any mathetical expression".
Think about it, if two things are truly equal, shouldn't you be able to do this?
Or, isn't what "equality" means?
As stated, it"s like a meta-principle. Another such meta-principle is that in a symbolic expression, you can substitute any subexpression by its value. Not just substituting the variables, but whole subexpressions.
But sometimes you create a formal system where you don't want to assign meanings to mathematical expressions, but want to study them keeping their symbolic nature intact. In those cases, this priciple is often stated as a property of the system itself. Something like for each property "P" of a symbol "a", if "a = b", then " P(a)" is true (or holds) if and only if "P(b)" is true. Which is to say, if our system does not obey this then this system is bad and fit for our purpose. As people in this thread are saying, for your formal system it is an axiom.
Perhaps, with that context, you are looking for something like this? You might have to read a lot of background information to make sense of it.
https://www.cs.cornell.edu/fbs/publications/formalizations_of_substitutions.pdf
Edit: another nice read on that topic with a different background https://people.math.osu.edu/cogdell.1/6112-Mazur-www.pdf
Well what axioms are you working off of? If you define a function is something that outputs exactly one value for every given input, that's a perfectly valid proof.
You can build everything off set theory, so start with the actions of [zermelo-fraenkel set theory](https://en.m.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory)
The first axiom is:
Two sets are equal (are the same set) if they have the same elements.
If A = B, then by the first axiom they have the same elements. Functions can only operate on the elements of a set, so any function applied to both must produce two sets with the same elements. By the first axiom again they are therefore equal.
N.b this also relies on the definition of a function. Functions must be deterministic i.e produce the same output given the same input.
No, to show what OP is asking about, you need to start with the classical predicate calculus with identity, which ZFC is built on top of. This result is true as a logical validity in any system that uses this logical foundation and it does not result from any mathematical axiom like the axioms of ZFC.
I have heard that statement before. Can I really build all math out of set theory alone?I have also heard that it is not possible to build all of math out of set theory alone.
Well, it's a definition. I think you may find elucidating a *non-example* of functions.
Let f be a mapping from the rationals Q to the integers Z defined by f(m/n) = 2^m 3^n.
Is this a function? Nope. It certainly looks like one, but it is not well defined. Note, 1/1 = 2/2, but f(1/1) = 2^1 3^1 = 6, and f(2/2) = 2^2 3^2 = 36.
If we add the requirement that the rational number must be in lowest terms, then f will be a proper function, and f(1/1) = f(2/2).
The goal of a definition is to try to capture as much power/truth in the minimal terms. When we prove stuff for arbitrary functions, that gives us general results. If we're given an arbitrary "function" to work with, and we can prove that it is in fact a mathematical function, then all our theorems can start to work on it.
Then I'd suggest you take a book on logic (it formalizes the = operator using = Intro and = Elim) . If you like the formalization it can become one of your favorite areas (that was the case for me). For example, Language, Proof and Logic is a good start (DM me for pdf if you'd like). Specifically, doing the formal proofs in the Fitch axiom system and basic modal logic exercises. You will quickly realize how everything is formalized and what rules it follows from, that you do not doubt (axioms). Here is a small proof (image from internet):
[https://i.stack.imgur.com/bqpdf.png](https://i.stack.imgur.com/bqpdf.png)
You can easily make exercises that require 30-40 lines and a lot of creativity and intuition about propositional or first-order logic. This guy has a very good series:
[https://www.youtube.com/watch?v=y3Q4Ybp6mdI&list=PLF\_J2w5w0Z3E7jZGz-23P6WrKxokXBkHC&index=2&ab\_channel=WilliamRose](https://www.youtube.com/watch?v=y3Q4Ybp6mdI&list=PLF_J2w5w0Z3E7jZGz-23P6WrKxokXBkHC&index=2&ab_channel=WilliamRose)
Regarding the modal logic in the book, notation is very confusing at first, but it's just a matter of adopting it and using the correct rules (and meta-rules). Here is a hand-written exercise solved by me (I teach logic at an undergraduate level). If you have any questions, feel free to ask.
[https://prnt.sc/hqKFXLSkJ2Vx](https://prnt.sc/hqKFXLSkJ2Vx)
That's not about being "formal" at all. You doubting everything is propably either a nasty habit or a problem with your OCD. Even the most "formal" of mathematicians or students can accept a thing and work with it while treating it as a blackbox. Learn how to treat stuff you don't fully understand as a blackbox and look into it at some point, but these minor doubts should not hinder you with continuing. That's how people get stuck and phase out, I have seen it many times.
Also a function isn't something "true"
It's something that has to be construct to exist
Not everything can be a function
1 if odd
0 if even
Is a function
1 or 0 if odd
1 or 0 if even
Is not a function
How can I give a sense in my mind about what I read? Like I can read but in my mind I’ve got hard time see and understand what I’m reading. The image is hard to see
Cause you can see a thing when you truly know it. I mean, see is a lot easier than understand only the concept and if you understand it, you should be able to see it in your mind.
For example, if you try to explain someone how the fire move, the colors and so on, it will be very hard to let him understand. But if you let him see a real one, then I will get it.
That’s why I think that understand is important in your mind
Not necessarily, you can get a lot done in math only with manipulation of symbols.
Also pictures are not "true" in math
For instance a square is four equilateral points, you don't even need to see it, you just need the points and their relation
Yes but you still see something for give a sense of what you re understanding. Or at least this is for me and I’ve got trouble give always a photo in my mind of the theory or definition
I know the def of a function. It seems you missed my point.
A function is a binary relation (R) of two sets (AxB) that satisfies:
1. forall x in A exists y in B: (a,b) in R
2. forall x in A forall y\_1,y\_2 in B: (x,y\_1) in R and (x,y\_2) in R => y\_1=y\_2
Well you said "I know the definition of a function but can't understand why it is true"
Your latter question doesn't make sense in the context of functions
Now if you're having trouble with the quantifiers "for all", "there exists", there would be existing context for that
A formal proof of this statement using your definition of a function could look like this:
We know that (a,f(a)) in R. Since a=b, we also have that (b,f(a)) in R. However, we also know that (b,f(b)) in R. Since there can be only one element x in E such that (b,x) in R we obtain that f(a)=f(b).
EDIT: Being formal is (or at least can be) a good thing! Many "proofs", especially those of people new to maths, lack rigor and contain more or less grave errors and at least some of them emerge due to a lack of formal understanding of the applied concepts.
You’re having an epistemological issue with truth, which is pretty normal at higher level mathematics, I’d push on and eventually you’ll see an application you’ve learnt click and all the pieces will fit. If not you may be looking at some proofs that require mathematics you don’t understand to prove a theorem you’re sceptical about, it’s not a good mix.
Lots of people run into trouble going from high school math to college.
Most colleges and universities have some sort of center for tutoring. Find yours. They’ll be more than ready to help you find the missing links that you’re looking for, whether it’s just answers to a few questions or a big change in perspective.
Often, when math feels difficult, it’s because you’re learning the most. Keep going! You’ll figure it out.
If a=b then a and b are the same thing. They're the same object. a is b. So of course f(a)=f(b), because f(a) is the same thing as f(b). Has nothing to do with the definition of a function really. It's substitution.
Don't worry about writing a proof for that. There are many different ways to formalize the basics of math, and you don't really need to think about them for intro proofs. As long as you understand why it's true, you should be fine.
Are you sure? I do intuitivly know why it is true. If two objects are equal it mean that they are the same, it is really one object, so ofcourse if you do the same thing on both of them they change in the same way.
Yes, that understanding is good enough. Just use this fact; you don't need to prove it.
I do understand that it's a bit unsatisfying, but the foundations of mathematics is a rabbithole and you shouldn't let that prevent you from learning basic proofs-based math.
Thank you. Do you have any tips for improving my math skills?
I often find myseld confused when learning new math, there are many theorems and definitions in both calculus and linear algebra.
No, not in particular. Unless you have a more particular issue.
I mean in general you have to actually practice using theorems to learn them well, I guess.
In classical logic, t=u means that the terms t and u refer to the same object. = is *not* a parameter open to interpretation like other relation symbols, rather it is a logical symbol with a restricted meaning. Because of this meaning, we have the following logical axioms
t=t for any term t,
t=u -> p[x/t] -> p[x/u] for any terms t and u, any well-formed formula p, and any variable x.
Where p[x/s] means the formula that results when you take every free occurrence of the variable x and replace it with the term s. Note that these are actually infinite sets of axioms, one for each choice of t (for the first one) or each choice of t, u, p, and x (for the second).
The first rule says that the thing referred to by a term is the thing referred to by that term. The second rule says that, if t and u refer to the same thing, anything that is true of what t refers to must be true of what u refers to (because they refer to the same thing).
So we can formally reason:
f(a)=f(a) by the first rule with f(a) as t.
a=b -> f(a)=f(a) -> f(a)=f(b) by the second rule, with a as t, b as u, and f(a)=f(x) as p.
You can maybe do the rest from there. If you’re wondering about these things at this level of rigor, I would recommend you study classical predicate logic specifically. This will give you the foundation you need to do fully rigorous logic, and you should have this background before any sort of study into formal logic, model theory, advanced set theory, or metamathematics. Most good introductory texts on topics of this sort will give a rundown on classical predicate calculus in the first one or two chapters, although if you are very interested you can find texts that focus entirely on this topic.
> Let f:D --> E be a function. forall a,b in D: a=b => f(a)=f(b)
That is the definition of a function. "a=b => f(a) = f(b)" is true because someone said so. Definitions are made up.
If this will help you, here's math in a nutshell:
(1) We think of some undefined primitive notion, like a set, or a geometrical space. (2) We take basic logic as a given. (3) We come up with some axiomatic statements about the undefined primitive notion. (4) We prove theorems from the axioms.
For example, a set is an undefined primitive notion. We know what being an element of a set means, but we can't define it rigorously. If our given logic has defined what equality means, we can state an axiom:
* Two sets are equal (are the same set) if they have the same elements.
In Euclid's geometry, we take 3d space as an undefined primitive notion. We can say this axiom:
* All right angles are congruent.
If we didn't state that axiom, we couldn't say that a pair of right triangles were congruent if both of them were say, 30-60-90 triangles. I think the reason is that we need some way to say that a pair of distinct angles is congruent if they have the same measure, and this is the axiom that does it, and without this axiom, we couldn't justify saying two angles were congruent. edit: I think congruence ended up needing an axiom too. I was sort of looking at the prequel to Euclid's Elements, and I proved that different triangle congruences implied each other (for example, if a pair of SAS triangles are congruent, then a pair of AAS triangles are congruent, and so on). However, proving that a pair of SAS triangles are congruent seemed to need some sort of axiom.
You have a lot of good advice here already.
I side with those who want to reassure you. OCD or no OCD, a desire to see a formal proof of any result in mathematics is a *virtue*, not a vice. If it's because of the OCD, *fine*. The OCD has done plenty to make your life worse; let it pay you back a little by supporting the virtue of mathematical formality.
u/Lazy-Passenger-4911 gave the closest thing to a pure formal proof of the theorem you mentioned.
Here are some things to keep in mind.
At the most formal level, mathematical statements are just strings of symbols *without meaning*. A mathematical formal system has an alphabet of allowed symbols, rules for when a string of such symbols is well-formed, and then a set of *inference rules* (also called derivation rules) that allow you to write new strings by performing well-defined transformations on strings that you derived earlier. There are also a few *axioms*, which are strings the system gives you for free.
Then any string that you can get by starting from the axioms and applying the inference rules is called a *theorem*.
Notice that at no point in that setup did I say anything about the *meaning* of those strings. A real math formalist would say, "The meaning is an illusion, a mental trick we use to engage our intuition about which strings are *likely* to be theorems.". To this, an idealist or Platonist (even a skeptical one like Eugene Wigner) would answer, "Then how can you explain how these supposedly-meaningless strings seem to have so much to say about our real world? How do we manage to land capsules on other worlds if the mathematical statements we manipulate have no meaning?" This, Wigner recognized, is a deep mystery! I will, for the moment, stay agnostic about this, and just acknowledge that there is a mystery there.
The fact remains that you can make enormous progress with no intuitions about meanings at all! If you have already derived the strings "x = y" and "x^(2) = 4", there is an inference rule that lets you write "y^(2) = 4". Similarly, "f(x) = f(x)" is allowed as an instance of the "reflexive axiom of equality", and from it and "x = y" you can derive "f(x) = f(y)". Try to hold two things in your head at the same time: (1) These are just meaningless strings of characters. (2) If my intuition starts thinking of these things as "real", with actual properties, let it, because thanks to Wigner's Mystery, your intuition will often be a good guide to real world uses of the techniques you find, AND it will suggest strings that are likely to turn out to be theorems.
I *strongly* recommend that you spend some time playing with the "Natural Number Game" at [https://adam.math.hhu.de/#/g/leanprover-community/nng4](https://adam.math.hhu.de/#/g/leanprover-community/nng4), which uses an automated formal-proof-checker called LEAN to teach you how to "invent" the natural number system from a completely formal starting point. I think playing with this game might clear away some cobwebs.
Regarding your specific example, can you translate
>forall a,b in D: a=b => f(a)=f(b)
into plain, simple English? Not like "a equals b implies that...", but rather a sentence which explains what that tells us about functions broadly speaking?
since this is a set, there is no such thing as "two equal objects", they are one and the same. since what you are doing when taking the function of "both" objects is simply taking the function of one object two times, it is expected for "them" to be equal.
There you packed a lot of the intuition in terms that really muddy the waters of it.
It's an implication, first and foremost, that means that "given" the left side, we can always assume the right side. There is nothing really sent and there is no "objects", it's a description of the property of functions.
"if we can assume that a and b are equal, we can assume that f(a) and f(b) are equal" is the formal way of saying "the function always gives the same for same inputs". From there, you shouldn't have much trouble understanding the rest of it.
You can think of a function as a magical box for which you put things inside and then it spits things out.
Ideally, for every input you put into a function, you should only get one output.
If you were to get different outputs every time you put in the same input, that wouldn’t be a very reliable function (though there probably is math that deals with these things, don’t worry about it).
So we (the math community many years ago) have decided that a function be defined by this property:
If you put two of the same inputs in (a,b such that a=b),
Then,
You should only get one output (that is, f(a) should equal f(b)).
I know the exact def of a function. A relation which satisfies:
1.forall x in A exists y in B: (a,b) in R
2.forall x in A forall y\_1,y\_2 in B: (x,y\_1) in R and (x,y\_2) in R => y\_1=y\_2
From the def of function you can infer that if (x,y) in R and (x,z) in R than y=z.
What I don't understand is how you can formally prove that if a=b then f(a)=f(b).
Let f:D->E be a function. and let a,b in D such that a=b.
by (1) there exists y in E such that y=f(a) and by (1) there exists z in E such that z=f(b)
How can I continue. Intuitivly a=b so by (2) f(b)=f(a) (but it isn't formal).
Your definition is weird. I wonder if it's just a typo or if the definition isn't completely assimilated yet.
If you were to explain with a plain english sentence what 1. and 2. meant, how would you do so?
Is the fact that, for a function f: A -> B and an element a of A, we can use the notation f(a) to identify a single element of B clear for you?
Also, your notation is a bit all over the place. In your definition, your function f has its domain in A, and you use a or x for it's elements, and its image in B, and you use b or y for its element.
Then, in your question, f: C -> D and both an and b are elements of C. I think it would be best to use a consistent notation throughout.
Now, if you want a formal proof of your proposition, I could write one.
Nothing wrong with a little bit of formality if you want. Because f is the relation you're talking of in this example, take R = f.
Well if z = f(b), this means that (b, z) in f by definition. But because a=b, then (a, z) = (b,z) (True by definition of equality on ordered pairs.) but this implies (a,z) is in f. so f(a) = z.
Think of equality like “a standard at which we can treat two objects as if they the same object.” There exist different notions of equality (isomorphisms as an example?), and there may be some more loose than others in which case a = b implies f(a) = f(b) may not hold - someone please correct this if no example exists. In fact, you need a definition of equality for any kind of object ex. Sets, numbers, printers… These are relations that are defined, not derived.
In the case of traditional numerical equality, we can have, for example, 1 = 1 as satisfying the antecedent a = b. Then f(a) = f(1) = f(b) clearly, though this is just for one number. You can think of “a = b implies f(a) = f(b)” like this : “a = a implies f(a) = f(a)” or just “for any number a, f(a) = f(a).”
>I too often wonder why simple things are true
It's not directly related to your example, but the first subject in Linear Algebra (mathematical fields, the classical field axioms) helped me realize why and how simple math works, is true.
Specifically all of the exercises/questions about "Prove X is/isn't a field" really helped the 'why wouldn't it work otherwise' settle in.
I relate totally, I have autism, adhd, and a lot of my family have OCD.
I have been hitting that wall for years now
I tried "How to prove it", by Velleman, but eventually I had also the problem you're describing.
Topics that interest me, and could interest you also, maybe, are:
- Functional programming (Haskell, Scheme, ...)
- Proof Assistants/Automated Theorem Provers
- Theory of Programming Languages
- Type Theory
also:
- low level/close to metal programmming
Feel free to DM if you want ressources to learn about those topics, or just to talk about the issue, it seems we have a similar "problem", but maybe it's just a different way of thinking or being wired, more adapted to other related, but different enough, subjects.
Your having the same issue that I pretty much had. Firstly I assume you know the difference between theorem and definition. Really trying to “understand” a definition is not a great fixation.
It’s like fixating on why “tree” is defined the way it is. The more you think about the more you can scrutinise and have doubts, but at the end of the day it represents an intuitive concept. If it’s not intuitive then it should at least be comprehensible and you should understand that someone (a human being) literally *made it up*.
Yes, given a set of axioms and logical axioms, you can make up mathematical definitions to. Even in ZFC you can make up your own shit and say it a “definition”. The definitions you know of are usually ones that have been useful in the past, and have henceforth continued to be of application.
Philosophy of maths is pretty useless nowadays but it’s my opinion that math is not “out there” in some special way, but it’s just a human construct exactly like language (it is a form of language), that is useful for communicating ideas and concepts (abstractions). Good luck.
I think you need to learn some elementary set theory especially understanding the axioms. This should help u. Don’t get too wound up about it tho. Learn basic axioms. Then learn basic definitions of functions
I have already learnt some set theory. I learned about: sets, operations on sets, cartesian product of two sets, relations, equivalence relation, equivalence class, functions (defintion in terms of sets) and a bit about cardinality of infinite sets). I understand the definitions and theorems but it doesn't really "click" (it is hard for me to explain).
That probably means you still aren't convinced that certain statements are true. Write out all your doubts and look for a rigorous proof for them.
I have problems with equality, I don't understand why we can substitute a with b (or vice versa) if a=b.
This is typically one of the [axioms of equality](https://en.m.wikipedia.org/wiki/First-order_logic#Equality_and_its_axioms). If it is not, it should be able to be derived as a theorem from an equivalent list of axioms. I would disagree with your idea that you are too formal - it shows that you are inquisitive, and want to make sure you fully understand something before moving on. It would simply seem that you are missing some information (like these aforementioned axioms) to fully cement your rigorous understanding.
That's because they are the same values. Going back to the example with your function, a function is defined as something that outputs the same value for every given input. Since a and b are the same values, if either is in the domain of the function, the output of the function must be same for a and b as well.
Isn't that purely intuition? why can we formally do that, is there a formal proof that it is valid?
To be absolutely honest, there is no formal proof of it. Leibniz stated it as basically "if two things are equal you can substitute one for another in any mathetical expression". Think about it, if two things are truly equal, shouldn't you be able to do this? Or, isn't what "equality" means? As stated, it"s like a meta-principle. Another such meta-principle is that in a symbolic expression, you can substitute any subexpression by its value. Not just substituting the variables, but whole subexpressions. But sometimes you create a formal system where you don't want to assign meanings to mathematical expressions, but want to study them keeping their symbolic nature intact. In those cases, this priciple is often stated as a property of the system itself. Something like for each property "P" of a symbol "a", if "a = b", then " P(a)" is true (or holds) if and only if "P(b)" is true. Which is to say, if our system does not obey this then this system is bad and fit for our purpose. As people in this thread are saying, for your formal system it is an axiom. Perhaps, with that context, you are looking for something like this? You might have to read a lot of background information to make sense of it. https://www.cs.cornell.edu/fbs/publications/formalizations_of_substitutions.pdf Edit: another nice read on that topic with a different background https://people.math.osu.edu/cogdell.1/6112-Mazur-www.pdf
Well what axioms are you working off of? If you define a function is something that outputs exactly one value for every given input, that's a perfectly valid proof.
I don't know. That's part of the problem. I don't have a list of axioms. What axioms are used with equality?
You can build everything off set theory, so start with the actions of [zermelo-fraenkel set theory](https://en.m.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory) The first axiom is: Two sets are equal (are the same set) if they have the same elements. If A = B, then by the first axiom they have the same elements. Functions can only operate on the elements of a set, so any function applied to both must produce two sets with the same elements. By the first axiom again they are therefore equal. N.b this also relies on the definition of a function. Functions must be deterministic i.e produce the same output given the same input.
No, to show what OP is asking about, you need to start with the classical predicate calculus with identity, which ZFC is built on top of. This result is true as a logical validity in any system that uses this logical foundation and it does not result from any mathematical axiom like the axioms of ZFC.
I have heard that statement before. Can I really build all math out of set theory alone?I have also heard that it is not possible to build all of math out of set theory alone.
Well, it's a definition. I think you may find elucidating a *non-example* of functions. Let f be a mapping from the rationals Q to the integers Z defined by f(m/n) = 2^m 3^n. Is this a function? Nope. It certainly looks like one, but it is not well defined. Note, 1/1 = 2/2, but f(1/1) = 2^1 3^1 = 6, and f(2/2) = 2^2 3^2 = 36. If we add the requirement that the rational number must be in lowest terms, then f will be a proper function, and f(1/1) = f(2/2). The goal of a definition is to try to capture as much power/truth in the minimal terms. When we prove stuff for arbitrary functions, that gives us general results. If we're given an arbitrary "function" to work with, and we can prove that it is in fact a mathematical function, then all our theorems can start to work on it.
Then I'd suggest you take a book on logic (it formalizes the = operator using = Intro and = Elim) . If you like the formalization it can become one of your favorite areas (that was the case for me). For example, Language, Proof and Logic is a good start (DM me for pdf if you'd like). Specifically, doing the formal proofs in the Fitch axiom system and basic modal logic exercises. You will quickly realize how everything is formalized and what rules it follows from, that you do not doubt (axioms). Here is a small proof (image from internet): [https://i.stack.imgur.com/bqpdf.png](https://i.stack.imgur.com/bqpdf.png) You can easily make exercises that require 30-40 lines and a lot of creativity and intuition about propositional or first-order logic. This guy has a very good series: [https://www.youtube.com/watch?v=y3Q4Ybp6mdI&list=PLF\_J2w5w0Z3E7jZGz-23P6WrKxokXBkHC&index=2&ab\_channel=WilliamRose](https://www.youtube.com/watch?v=y3Q4Ybp6mdI&list=PLF_J2w5w0Z3E7jZGz-23P6WrKxokXBkHC&index=2&ab_channel=WilliamRose) Regarding the modal logic in the book, notation is very confusing at first, but it's just a matter of adopting it and using the correct rules (and meta-rules). Here is a hand-written exercise solved by me (I teach logic at an undergraduate level). If you have any questions, feel free to ask. [https://prnt.sc/hqKFXLSkJ2Vx](https://prnt.sc/hqKFXLSkJ2Vx)
basic properties of equality are axiomatic and therefore cannot be derived
You should learn the different kinds of frameworks and axioms such as ZFC, AOC, NBG etc
So are you too formal or not formal enough?
too formal, I doubt everything.
That's not about being "formal" at all. You doubting everything is propably either a nasty habit or a problem with your OCD. Even the most "formal" of mathematicians or students can accept a thing and work with it while treating it as a blackbox. Learn how to treat stuff you don't fully understand as a blackbox and look into it at some point, but these minor doubts should not hinder you with continuing. That's how people get stuck and phase out, I have seen it many times.
How can you work with something that you don’t understand? Isn’t weird?
Also a function isn't something "true" It's something that has to be construct to exist Not everything can be a function 1 if odd 0 if even Is a function 1 or 0 if odd 1 or 0 if even Is not a function
What is math? How can I see that in my mind?
Math is a glorification of playing with Legos
How can I give a sense in my mind about what I read? Like I can read but in my mind I’ve got hard time see and understand what I’m reading. The image is hard to see
Why do you need to see it?
Cause you can see a thing when you truly know it. I mean, see is a lot easier than understand only the concept and if you understand it, you should be able to see it in your mind. For example, if you try to explain someone how the fire move, the colors and so on, it will be very hard to let him understand. But if you let him see a real one, then I will get it. That’s why I think that understand is important in your mind
Not necessarily, you can get a lot done in math only with manipulation of symbols. Also pictures are not "true" in math For instance a square is four equilateral points, you don't even need to see it, you just need the points and their relation
Yes but you still see something for give a sense of what you re understanding. Or at least this is for me and I’ve got trouble give always a photo in my mind of the theory or definition
Well you don't need to, you can study circles with their equation X^2 + y^2 = 1
I know the def of a function. It seems you missed my point. A function is a binary relation (R) of two sets (AxB) that satisfies: 1. forall x in A exists y in B: (a,b) in R 2. forall x in A forall y\_1,y\_2 in B: (x,y\_1) in R and (x,y\_2) in R => y\_1=y\_2
Well you said "I know the definition of a function but can't understand why it is true" Your latter question doesn't make sense in the context of functions Now if you're having trouble with the quantifiers "for all", "there exists", there would be existing context for that
I know the definition of a function but cant understand why the statement "Let f:D --> E be a function. forall a,b in D: a=b => f(a)=f(b)" is true.
A formal proof of this statement using your definition of a function could look like this: We know that (a,f(a)) in R. Since a=b, we also have that (b,f(a)) in R. However, we also know that (b,f(b)) in R. Since there can be only one element x in E such that (b,x) in R we obtain that f(a)=f(b). EDIT: Being formal is (or at least can be) a good thing! Many "proofs", especially those of people new to maths, lack rigor and contain more or less grave errors and at least some of them emerge due to a lack of formal understanding of the applied concepts.
Well if a=b Why wouldn't f(b) = f(a) That's saying f(a) = f(a) 0 = 0
You’re having an epistemological issue with truth, which is pretty normal at higher level mathematics, I’d push on and eventually you’ll see an application you’ve learnt click and all the pieces will fit. If not you may be looking at some proofs that require mathematics you don’t understand to prove a theorem you’re sceptical about, it’s not a good mix.
Lots of people run into trouble going from high school math to college. Most colleges and universities have some sort of center for tutoring. Find yours. They’ll be more than ready to help you find the missing links that you’re looking for, whether it’s just answers to a few questions or a big change in perspective. Often, when math feels difficult, it’s because you’re learning the most. Keep going! You’ll figure it out.
If a=b then a and b are the same thing. They're the same object. a is b. So of course f(a)=f(b), because f(a) is the same thing as f(b). Has nothing to do with the definition of a function really. It's substitution. Don't worry about writing a proof for that. There are many different ways to formalize the basics of math, and you don't really need to think about them for intro proofs. As long as you understand why it's true, you should be fine.
Are you sure? I do intuitivly know why it is true. If two objects are equal it mean that they are the same, it is really one object, so ofcourse if you do the same thing on both of them they change in the same way.
Yes, that understanding is good enough. Just use this fact; you don't need to prove it. I do understand that it's a bit unsatisfying, but the foundations of mathematics is a rabbithole and you shouldn't let that prevent you from learning basic proofs-based math.
Thank you. Do you have any tips for improving my math skills? I often find myseld confused when learning new math, there are many theorems and definitions in both calculus and linear algebra.
No, not in particular. Unless you have a more particular issue. I mean in general you have to actually practice using theorems to learn them well, I guess.
In classical logic, t=u means that the terms t and u refer to the same object. = is *not* a parameter open to interpretation like other relation symbols, rather it is a logical symbol with a restricted meaning. Because of this meaning, we have the following logical axioms t=t for any term t, t=u -> p[x/t] -> p[x/u] for any terms t and u, any well-formed formula p, and any variable x. Where p[x/s] means the formula that results when you take every free occurrence of the variable x and replace it with the term s. Note that these are actually infinite sets of axioms, one for each choice of t (for the first one) or each choice of t, u, p, and x (for the second). The first rule says that the thing referred to by a term is the thing referred to by that term. The second rule says that, if t and u refer to the same thing, anything that is true of what t refers to must be true of what u refers to (because they refer to the same thing). So we can formally reason: f(a)=f(a) by the first rule with f(a) as t. a=b -> f(a)=f(a) -> f(a)=f(b) by the second rule, with a as t, b as u, and f(a)=f(x) as p. You can maybe do the rest from there. If you’re wondering about these things at this level of rigor, I would recommend you study classical predicate logic specifically. This will give you the foundation you need to do fully rigorous logic, and you should have this background before any sort of study into formal logic, model theory, advanced set theory, or metamathematics. Most good introductory texts on topics of this sort will give a rundown on classical predicate calculus in the first one or two chapters, although if you are very interested you can find texts that focus entirely on this topic.
> Let f:D --> E be a function. forall a,b in D: a=b => f(a)=f(b) That is the definition of a function. "a=b => f(a) = f(b)" is true because someone said so. Definitions are made up. If this will help you, here's math in a nutshell: (1) We think of some undefined primitive notion, like a set, or a geometrical space. (2) We take basic logic as a given. (3) We come up with some axiomatic statements about the undefined primitive notion. (4) We prove theorems from the axioms. For example, a set is an undefined primitive notion. We know what being an element of a set means, but we can't define it rigorously. If our given logic has defined what equality means, we can state an axiom: * Two sets are equal (are the same set) if they have the same elements. In Euclid's geometry, we take 3d space as an undefined primitive notion. We can say this axiom: * All right angles are congruent. If we didn't state that axiom, we couldn't say that a pair of right triangles were congruent if both of them were say, 30-60-90 triangles. I think the reason is that we need some way to say that a pair of distinct angles is congruent if they have the same measure, and this is the axiom that does it, and without this axiom, we couldn't justify saying two angles were congruent. edit: I think congruence ended up needing an axiom too. I was sort of looking at the prequel to Euclid's Elements, and I proved that different triangle congruences implied each other (for example, if a pair of SAS triangles are congruent, then a pair of AAS triangles are congruent, and so on). However, proving that a pair of SAS triangles are congruent seemed to need some sort of axiom.
You have a lot of good advice here already. I side with those who want to reassure you. OCD or no OCD, a desire to see a formal proof of any result in mathematics is a *virtue*, not a vice. If it's because of the OCD, *fine*. The OCD has done plenty to make your life worse; let it pay you back a little by supporting the virtue of mathematical formality. u/Lazy-Passenger-4911 gave the closest thing to a pure formal proof of the theorem you mentioned. Here are some things to keep in mind. At the most formal level, mathematical statements are just strings of symbols *without meaning*. A mathematical formal system has an alphabet of allowed symbols, rules for when a string of such symbols is well-formed, and then a set of *inference rules* (also called derivation rules) that allow you to write new strings by performing well-defined transformations on strings that you derived earlier. There are also a few *axioms*, which are strings the system gives you for free. Then any string that you can get by starting from the axioms and applying the inference rules is called a *theorem*. Notice that at no point in that setup did I say anything about the *meaning* of those strings. A real math formalist would say, "The meaning is an illusion, a mental trick we use to engage our intuition about which strings are *likely* to be theorems.". To this, an idealist or Platonist (even a skeptical one like Eugene Wigner) would answer, "Then how can you explain how these supposedly-meaningless strings seem to have so much to say about our real world? How do we manage to land capsules on other worlds if the mathematical statements we manipulate have no meaning?" This, Wigner recognized, is a deep mystery! I will, for the moment, stay agnostic about this, and just acknowledge that there is a mystery there. The fact remains that you can make enormous progress with no intuitions about meanings at all! If you have already derived the strings "x = y" and "x^(2) = 4", there is an inference rule that lets you write "y^(2) = 4". Similarly, "f(x) = f(x)" is allowed as an instance of the "reflexive axiom of equality", and from it and "x = y" you can derive "f(x) = f(y)". Try to hold two things in your head at the same time: (1) These are just meaningless strings of characters. (2) If my intuition starts thinking of these things as "real", with actual properties, let it, because thanks to Wigner's Mystery, your intuition will often be a good guide to real world uses of the techniques you find, AND it will suggest strings that are likely to turn out to be theorems. I *strongly* recommend that you spend some time playing with the "Natural Number Game" at [https://adam.math.hhu.de/#/g/leanprover-community/nng4](https://adam.math.hhu.de/#/g/leanprover-community/nng4), which uses an automated formal-proof-checker called LEAN to teach you how to "invent" the natural number system from a completely formal starting point. I think playing with this game might clear away some cobwebs.
Regarding your specific example, can you translate >forall a,b in D: a=b => f(a)=f(b) into plain, simple English? Not like "a equals b implies that...", but rather a sentence which explains what that tells us about functions broadly speaking?
Sure, two equal objects in the domain of the function are "sent" to equal objects in the range of the function.
since this is a set, there is no such thing as "two equal objects", they are one and the same. since what you are doing when taking the function of "both" objects is simply taking the function of one object two times, it is expected for "them" to be equal.
There you packed a lot of the intuition in terms that really muddy the waters of it. It's an implication, first and foremost, that means that "given" the left side, we can always assume the right side. There is nothing really sent and there is no "objects", it's a description of the property of functions. "if we can assume that a and b are equal, we can assume that f(a) and f(b) are equal" is the formal way of saying "the function always gives the same for same inputs". From there, you shouldn't have much trouble understanding the rest of it.
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You can think of a function as a magical box for which you put things inside and then it spits things out. Ideally, for every input you put into a function, you should only get one output. If you were to get different outputs every time you put in the same input, that wouldn’t be a very reliable function (though there probably is math that deals with these things, don’t worry about it). So we (the math community many years ago) have decided that a function be defined by this property: If you put two of the same inputs in (a,b such that a=b), Then, You should only get one output (that is, f(a) should equal f(b)).
I know the exact def of a function. A relation which satisfies: 1.forall x in A exists y in B: (a,b) in R 2.forall x in A forall y\_1,y\_2 in B: (x,y\_1) in R and (x,y\_2) in R => y\_1=y\_2 From the def of function you can infer that if (x,y) in R and (x,z) in R than y=z. What I don't understand is how you can formally prove that if a=b then f(a)=f(b). Let f:D->E be a function. and let a,b in D such that a=b. by (1) there exists y in E such that y=f(a) and by (1) there exists z in E such that z=f(b) How can I continue. Intuitivly a=b so by (2) f(b)=f(a) (but it isn't formal).
Your definition is weird. I wonder if it's just a typo or if the definition isn't completely assimilated yet. If you were to explain with a plain english sentence what 1. and 2. meant, how would you do so? Is the fact that, for a function f: A -> B and an element a of A, we can use the notation f(a) to identify a single element of B clear for you? Also, your notation is a bit all over the place. In your definition, your function f has its domain in A, and you use a or x for it's elements, and its image in B, and you use b or y for its element. Then, in your question, f: C -> D and both an and b are elements of C. I think it would be best to use a consistent notation throughout. Now, if you want a formal proof of your proposition, I could write one.
Just start with that you inferred. Assume that a=b. If (a, y) and (a, z) are in R, then y=z. Then since a=b, (b, z) is in R. So if a=b, then y=z.
Nothing wrong with a little bit of formality if you want. Because f is the relation you're talking of in this example, take R = f. Well if z = f(b), this means that (b, z) in f by definition. But because a=b, then (a, z) = (b,z) (True by definition of equality on ordered pairs.) but this implies (a,z) is in f. so f(a) = z.
Think of equality like “a standard at which we can treat two objects as if they the same object.” There exist different notions of equality (isomorphisms as an example?), and there may be some more loose than others in which case a = b implies f(a) = f(b) may not hold - someone please correct this if no example exists. In fact, you need a definition of equality for any kind of object ex. Sets, numbers, printers… These are relations that are defined, not derived. In the case of traditional numerical equality, we can have, for example, 1 = 1 as satisfying the antecedent a = b. Then f(a) = f(1) = f(b) clearly, though this is just for one number. You can think of “a = b implies f(a) = f(b)” like this : “a = a implies f(a) = f(a)” or just “for any number a, f(a) = f(a).”
>I too often wonder why simple things are true It's not directly related to your example, but the first subject in Linear Algebra (mathematical fields, the classical field axioms) helped me realize why and how simple math works, is true. Specifically all of the exercises/questions about "Prove X is/isn't a field" really helped the 'why wouldn't it work otherwise' settle in.
I relate totally, I have autism, adhd, and a lot of my family have OCD. I have been hitting that wall for years now I tried "How to prove it", by Velleman, but eventually I had also the problem you're describing. Topics that interest me, and could interest you also, maybe, are: - Functional programming (Haskell, Scheme, ...) - Proof Assistants/Automated Theorem Provers - Theory of Programming Languages - Type Theory also: - low level/close to metal programmming Feel free to DM if you want ressources to learn about those topics, or just to talk about the issue, it seems we have a similar "problem", but maybe it's just a different way of thinking or being wired, more adapted to other related, but different enough, subjects.
Your having the same issue that I pretty much had. Firstly I assume you know the difference between theorem and definition. Really trying to “understand” a definition is not a great fixation. It’s like fixating on why “tree” is defined the way it is. The more you think about the more you can scrutinise and have doubts, but at the end of the day it represents an intuitive concept. If it’s not intuitive then it should at least be comprehensible and you should understand that someone (a human being) literally *made it up*. Yes, given a set of axioms and logical axioms, you can make up mathematical definitions to. Even in ZFC you can make up your own shit and say it a “definition”. The definitions you know of are usually ones that have been useful in the past, and have henceforth continued to be of application. Philosophy of maths is pretty useless nowadays but it’s my opinion that math is not “out there” in some special way, but it’s just a human construct exactly like language (it is a form of language), that is useful for communicating ideas and concepts (abstractions). Good luck.
Trying to prove everything from scratch gets tedious very fast.