Aren’t you using logic to prove logic to be true? And isn’t that circular? I personally have my doubts over this true and false dichotomy through the observation of our universe. Surely, logic is a man-made creation that uses man-made definitions to describe events or object’s behavior. Indeed, it seems to be working well for day to day activities and has advanced us in society. But to say that something is true or that something is false is no different than saying that something is hot or cold, light or dark, positive or negative, matter or space. These can be vague depending on how you define it, but the definitions I give them below I think are precise.

The examples I gave above do not truly exist in this universe, because one is the absence of the other. For one to exist, the other has to cease to exist. If we consider one degree above absolute zero as “heat”, or if we were to define one photon of light as “light”, or one proton as “positive”, then what we measure as cold, dark, and negative are nothing more than the absence of heat, photons, or protons. It seems to be that we have given the negation a definition when there is nothing there to define. Can nothing be defined?

The space or matter definition needs to be defined on its own. One cannot have matter without it being contained in space itself. For example, a lake requires the rocks and the sand underneath it to exist; it couldn’t just float in the air. Even if it did float in the air, it requires the space in the air to float. In other words, matter needs a placeholder or container to exist.

So there seems to be two values, hot vs cold, light vs dark, positive vs negative, matter vs space, but since one is the absence of the other, only one true value really exists. In one of your talks, “The Importance of Mathematics”, you give a presentation of knots. The knot at the very bottom of the slide represents what I am talking about in this post (and is my profile picture, the infinity symbol). It seems like there are two sides and two circles in the equation, but if you untwist the infinity symbol, or the symbol you had in your presentation, you get one circle. Therefore it seems as if these are two separate identities in our universe, but they are all just one in reality.

I am very interested in this discussion, and if you can point me in the right direction, or if you can provide me with some writing or explanation of how the points I made here are flawed or supported, I would greatly appreciate it.

Having said all that, I really appreciate the time you have put aside to educate people in writing. I am a strong supporter of this movement and I applaud your dedication towards the study of mathematics.

Thank you again,

Nima Farzaneh

Well, for a start, showing that a is the smallest element of A is not sufficient to show that it is not the largest.

]]>In the example where ‘a is not the largest element of A’, we could show that a is the least element of A or that there exists b and c such that b is larger than a and c is larger than a. Don’t these statements still prove that a is not the largest element of A? ]]>

I have a question. It seems that the falsehood of a given statement is being defined in terms of the truth of another statement. How do we define truth? If we defined truth in terms of falsehood is there any circularity?

Thanks.

]]>I’m not entirely sure, but I think there exist some authors who don’t think an “if” in a definition really means “iff” (I don’t understand how they think this, and I don’t recall where I read this). Also, if definitions have “iffs” in them, then one might claim that anywhere we can have an “iff” we can have a definition (as I understand them, the Polish school of logicians around the 1920s held this position). This would allow for very strange definitions as permissible, like in classical propositional logic defining the “weak law of identity” as “hypothetical syllogism” or any other theorem of classical propositional logic, e. g. I could define Cpp as CCpqCCqrCpr or CqCpq or any other theorem. Perhaps that isn’t a good criticism, perhaps very strange definitions should come as permissible. And perhaps that isn’t why “iff” gets avoided in definitions, but perhaps it does.

]]>“A statement is taken as true if there is a method for demonstrating it.”

Hmm, can I say that this is more or less a matter about how one deals with INFINITE? As in the Peano axioms, mathematical induction is one of the axioms, which I think, is not accepted by the “intuitionistic mathematics”.

You have some similar examples, but ordinary usage is sometimes just sloppiness. When double negatives are used in court, it is usually an attempt to use the words precisely.

]]>Even outside explicit ‘Definitions’, definitions are usually flagged in a couple of ways:

1. In textbooks, terms being defined are generally in italics, bold or both. Many lecturers underline terms being defined in the same way.

2. A large proportion of definitions use one of six or so special ‘defining’ verbs, such as ‘call’. (E.g. ‘We call f an injection if …’.) These verbs unambiguously tell you that you are dealing with a definition instead of an assertion.

So there’s a little room for confusion, but not very much.

]]>I think I’ll add a remark at the beginning of the first post, to the effect that the length of the posts is supposed to be offset by the ease with which they can be read.

]]>Summaries and minimizing the need to backtrack seem like very good ways to help avoid the issues we’re talking about.

]]>@gowers I agree. Which includes admitting that my remark was not helpful in any practical way. I don’t know what’s best either.

@ Aspirant mathmo As you point out, the material won’t be too far removed from what the average incoming student already sort-of knows (though maybe just not presented in this style, like you say). However, despite it being difficult for me to imagine not having heard of Professor Gowers and his blog, I think even at Cambridge it might be a bit too optimisitc to suppose that the `average’ incoming student reads Gowers’s blog. One might even go so far as to argue that it is the less obviously ambitious students, i.e. those who aren’t already doing keen things like reading maths blogs that will need this kind of extra support the most.

I don’t really know the answer, but I’ll at least make sure the first years at my college hear about these posts sooner rather than later in order to minimize the possible negative effect I conjectured. Regardless of how you choose to continue, I very much look forward to future posts. Many Thanks.

]]>That’s quite reassuring. The intention was that it should be possible to read the posts without continually having to stop and think, or, worse, backtrack. If I’ve done what I wanted, then although the posts are quite long, they are also quicker to read than a much shorter segment of a textbook. Also, I have written a summing-up post that’s much shorter, so I’m hoping that people will read the longer posts fairly quickly, but not feel that they have to hold everything in their heads straight away (though eventually this is all stuff a mathematician needs to know). Rather, they can get the general idea from the longer posts and refer back to the summary if they need reminding of the key points. Or some people might prefer to read the summary and then refer to the posts if they find parts of the summary that they are not comfortable with.

I’ll also repeat that if anyone reading this knows other freshers who might not know about it, I’d be grateful if you could spread the word.

]]>As an incoming Cambridge mathematics fresher, I’d say that much of what Professor Gowers has currently posted on basic logic is material many new undergraduates are likely to be familiar with, yet may not have encountered formal definitions/ explanations of. Although having a long list of posts may seem intimidating, the actual content of the introductory logic posts is not especially challenging if it has been encountered informally beforehand (yet it is still extremely worthwhile reading). My understanding of the first post in the series was that the logic posts are something of a “warm-up” for the rest of the blog… Given that much of this material may otherwise go assumed/unclarified by lecturers, I think the current approach of a post every couple of days on basic logic is a good way to go. That way, most of the basic logic can be dealt with before we start meeting more juicy material in lectures. I don’t know how many freshers will actually be reading this blog before lectures start though. Many thanks to Professor Gowers for the blog so far: I’m looking forward to reading more!

]]>I’m not quite sure what to think about that. On one side is the argument you give, while on the other is the argument that these posts are preliminary, and therefore best understood *before* term starts. I genuinely don’t know what the best policy is.

Thanks very much for that link, which I’ll recommend in a later post. I had been considering inventing some questions like that myself, but these are perfect and save me the bother. (I remember Terence Tao’s blog post about multiple choice questions but I’d completely forgotten that there was an applet.

]]>Actually, I stand by what I wrote in that section, except that it should not be taken as a description of intuitionism. That is, it is genuinely the case that the law of the excluded middle is not obviously appropriate for vague statements, and it is genuinely not clear to a non-Platonist like me that every well-formed mathematical statement should have a determinate truth value. What I might do is rewrite the passage to make it clearer that by expressing these views I am not describing intuitionism.

]]>The worry about using “if” to mean “iff” is that one won’t recognize an intended definition as a definition. If the whole context is prefaced by the word “Definition”, as it often is in math papers, then Prof. Tao’s justification works. If not, it doesn’t necessarily work.

]]>One of the categories is Logic. If you are a student who is not sure if you already know the material in these posts here on logic then it might be worthwhile to try the 16 multiple choice logic questions on Terence Tao’s web and see how you do.

Conversely if you know you did not know the material in these posts but now after reading them you think you do it might be worthwhile to test yourself and see if you do!

]]>I’d like to add a little motivation to this (by no means the only motivation): computer algorithms are fundamentally finitary. There is a huge field of computational geometry which pretty much only has an existence because classical geometry assumes things like comparability of reals — whereas in reality it’s simply uncomputable whether two arbitrary reals are the same (note that it’s possible to be sure they’re different!).

]]>Otherwise, I’m enjoying this series!

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