This book was a good self learning intro to proofs for me. When I went back to school for a math degree and my undergrad analysis professor told me I am already studying like a grad student I took that as a huge compliment.
Instead of "unicorns" in your proofs, you may want to say "observed equine unicorns". Unicorn used to refer to some rhinos, and given the apparently infinite size of our universe, it is more probable that classic "equine" unicorns really exist out there than not. I cannot attest to their purpleness or math skills.
By that logic, wouldn't it be the case that both statements: Unicorns are purple. and Unicorns are not purple [/Unicorns are of colors other than purple]. ...are both [vacuously] true? Honestly, the idea of vacuous truth doesn't make sense to me (does it have any practical use, or is it just a convention?). My main issue is that every statement is sort of a hypothesis about the world and its properties. And some statements have hidden premises. If someone says that Unicorns are purple, they presume that Unicorns exist. Thus it's really: Unicorns exist & All Unicorns are purple. Which ends up being false, or maybe just meaningless. Surely it's not true.
Also wouldn't the statement: Guns don't kill people; people do - be vacuously true. Obviously you don't see guns going off by themselves. It takes a person to press the trigger to release the bullet. The point is the statement is meaningless and it's a really inane argument to try to argue for no gun restrictions. The fact of the matter is guns are the most dangerous "tools" ever invented especially assault rifles. And furthermore on the logic part, you don't see cars driving themselves. (not yet) it takes a person behind the wheel to drive a car. It takes a person to bake a cake, it takes a person to write a book. So in conclusion just to reiterate, the statement that people kill people is a really dumb and vapid argument on the Republican side.
A bit late but this is one of my favorite books that I even travel with. A lot of times I have to prove stuff and if I'm stuck, I reread the relevant section, follow the advice given (reread the theorem, write the definitions, etc) and usually I end up getting the proof I need. Excellent recommendation!
Velleman is incredibly thorough and well laid out - which can mean it takes effort to work through its content. There are several free, excellent online books, particularly Hammack Book of Proof, that complement Velleman. However, if I want a definitive answer on how to analyze a particular proposition and how to devise a strategy to tackle the proof, I turn to Velleman.
I have that book (same edition). It confused me because some of the terms did not match my bridge class. I will give it another look now. Thanks. Would a lesson on the biconditional truth table clarity your logic lesson in a more visual manner?
Statements like x^2 >= x implies x =1 always interested me when I was first studying math. Because the RHS gives a potentially larger set of values in which the LHS is true. Furthermore, the statement could be vacuous! But since the reverse direction is of course true not only is it not vacuous, but that the set of values the LHS is true is precisely given by the RHS. I think it's important to enjoy thinking deeply about things such as this if anyone wants to become a math major
Nice video. Based on this I ordered two books, this one and “Book of Proof” by Hammack. Hammack has odd numbered worked problems and a better rating on Amazon. At Auburn we’re we’re taught using the “Texas Method” i.e. no books, Math Prof put a few axioms on the board along with some theorems to prove for the next class. In the next class, folks were called to show their proofs and typically get shot down (except for the math brainiacs who always sat in the back rows). Then more theorems were added for the next class and so on. Problem is, no books like these too go back to and review. So you better have some serious notes to get thru the theorems on the final exam, which was 100% of the class grade, usually. The whole math department was run like this. I have 40+ quarter hours of theoretical math as an applied mathematician. Wretched! But, as a Military Operations Research Analyst it didn’t matter as you had to learn on the job as there were no classes and no textbooks anyway back in the 70s anyway. The good thing was, the Texas Method taught you to think logically, under pressure in from of a bunch of snipers…and in hindsight, I’m the better for it as it turned out.
Love it. I find it also related with philosophy. The reason why I related to philosophy at some point is when writing some good arguments. Even though you are not writing arguments in math (or at least not that specifically), I find it quite similar, since you need also premises to make it makes sense for writing math proofs as well. Thank you for this video
so because we cannot concretely quantify or attach a value, action or law to basis set- unicorns this pretty much leads to infinite possibilities of statements, which we cannot prove false, then they must be true?
I know what a math statement is..but am unsure if unicorns have to be either purple or not if they don’t exist, e.i. is this really a math statement? The part about all unicorns being purple seems like it slightly depends on semantics and what’s considered the convention for how to interpret this (specifically it seems to rely on two ideas: 1: a statement either is true or false; therefore, if a statement is not true it is false, vice versa); 2: “all unicorns are purple” is a statement”. I have trouble accepting the second idea. I think I understand that if “there is a unicorn that’s not purple” is false, then the negation of said statement is true, i.e, “it’s not the case that there is a unicorn that’s not purple”. **This true statement (at least mathematically) means “for any thing that’s a unicorn, it’s purple”or “all unicorns are purple”).** THIS is where my philosophy brain has an issue, only the last premise in bold. I don’t see why this definitional convention that a statement is either true or false must extend to descriptions of things that don’t exist in reality. Cannot the claim “all unicorns are purple” be neither true or false (by a different semantic convention of what constitutes a claim)? Or perhaps, from a yet another semantic approach, can it not be considered that a descriptive statement z about something x is only true or false if x has defined parameters for what it is and z describes properties that are either consistent or inconsistent with said parameters of x? By that new convention, “all unicorns are purple” could neither be confirmed or denied, neither true or false, since unicorns are not constrained to be either purple or not purple (not by arbitrary definition nor by observable reality). In other words, unicorns don’t have parameters z that pertain to the description x of being or not being purple; therefore, “all unicorns are purple” is not true or false since there are no unicorns to which the description purple/not purple is attributable, nor is color an integral component of the definition of unicorn. Maybe I shouldn’t bring philosophy into this? Maybe the true/false dichotomy for descriptions of nonexistent things is simply more useful to us, but not logically necessary? Idk mayne!
I'm familiar with Velleman through his philosophical work, so please being philosophy into this. You seem to know what you're talking about, so for anyone that reads this and is interested further, look up Quine and truth and unicorns. Also perhaps study some of Leibniz writings on infinity and logical possibilty
This reminds me, and looks a lot like the frame of arguments (both inductive and deductive) taught in philosophy modules where you have; premise one, premise two and then a conclusion. (E.g. All swans are white, this is a swan, therefore this is white. If that's the case, is the mathematical notion of (vacuously) true the same as the philosophical is/exist? And I just looked up vacuously, I found that it means mindlessly or blankly etc. If that is what is meant here, is vacuously true actually just meaningless? Is it just something that we hold in our heads so we can entertain the learning point?
Yeah, "vacuous truth" seems to me to be a misnomer. The statement is either meaningless, or outright false. Every statement is a hypothesis about the nature of the world or its parts and most of the time you should be able to represent it as a conjunction of more basic statements. For instance, "The Earth is a 3rd planet from the Sun" assumes that the Sun exists, that there are some constraints as to which astronomical bodies count as planets, that Earth is a planet and that exactly two other planets are closer to the Sun than Earth (arguably you could include many many further, more basic statements there). In the same way, "Unicorns are purple" is a hypothesis that could be represented at minimum by the conjunction: "Unicorns exist" & "Unicorns are purple". The first part is false. Whether the second part is interpreted as false, [vacuously] true, or meaningless, the end result should be either false or meaningless anyway.
Hmm, they are both excellent books. They have different examples so I think it's worth having both. Note this one is a softcover but I do like it a lot.
Hey math sorcerer, I have not started mathematical logic, should I get a book that talks about it before getting this one or do they also teach it in parallel to proof writing
@@TheMathSorcerer So for a universty student of Biology, do you recommend this book? Because I don't know if my level of Maths is in part with the book
as a HS student who is still struggling with Algebra and Calculus, but does lots of programming which need a lot of logic Proof is kinda like Programming, but in different language. Only shapes are used
I'm going through Susanna Epp's "Discrete Mathematics with Applications" and I totally see why you had it recommended as a starting book. Although I haven't gotten far into the book, I can see how the beginning chapter on logic definitely bleeds into all of mathematics!
This is a book I highly recommend. It can be a grind for some of the proofs if you do not have any help from a classroom or mentor/tutor since there are no solutions. That being said there are many of the problems that are not difficult and you can build on those to help with those problems that you have difficulty with. Good suggestion. I am going through the free book recommend from another of your videos right now just to see if it brings something different. Attacking from different angles and books helps to illuminate the path to a solution. Good luck.
Another way of saying that anyone who exists in a universe without The Math Sorcerer is great at math - as long as no other universe exists. If one does, its members lose that cool quality, because they never had both it and existence.
I suppose it depends on whether you define 'real' as 'existing'. If so, I think the statement would be false by definition. E(x) = ¬(x∈∅) A = {all unicorns} = ∅ ∀ x∈A : x∈A x∈∅ ¬E(x)
Here is a statement: All unicorns have 3 eyes. Is it true? If it is proved to be mathematically true by following the same procedure, that's weird and upsetting. I know the underlying text, unicorns do not exist physically, the subject of the statement is void vacuum, or its existence cannot be verified.
k o . b -ok . a si a /s/h ow% 20to%20prov e% 20i t go to that link i put them apart because over a month ago i put a link or 2 or a lot in the comments section and youtube deleted it...
