That's a historical artifcat, not an innate property. (Math was first invented in the Bronze Age for accounting and architecture.)
There's a theoretical parallel universe with a mathematics where numbers aren't first on the ladder but down there around number five. (Ironically computers are an example of that. Figuring out how to do numbers when all you have is finite sequences from an alphabet is advanced CS.)
Firstly are you so very sure he claimed that is an innate property? That numbers are a gateway to quantitative reasoning seems a perfectly reasonable claim either way.
Secondly there are lots of mathematical objects that significantly predate the bronze age (eg https://en.wikipedia.org/wiki/Ishango_bone but there are others) and because these objects are so old we have no idea why math was first invented.
Thirdly I have no idea the relevance of the theoretical “late numbers “ parallel universe but this thread is about a book in this one.
> That Numbers are a gateway to quantitative reasoning seems a perfectly reasonable claim either way
This isn't true though. It's not supported by anything except Bronze Age tradition.
I.e., when teaching kids advanced math at an early age it's often better to skip numbers altogether and start with set theory or logic. Many kids find that a much easier gateway.
Set theory and logic stay below numbers, but they do not belong to quantitative reasoning.
By definition, quantities, either discrete or continuous, as already classified by Aristotle, are the properties that can be expressed by numeric values, as a result of counting for discrete quantities, or as a result of measuring for continuous quantities.
Properties that are not quantities, but which can be compared, to determine which is greater or less, or if they are equal, are qualities.
The simplest numbers, i.e. cardinal numbers a.k.a. natural numbers, are equivalence classes of sets, so indeed the theory of numbers is built upon that of sets (a.k.a. multitudes in older language), but this does not contradict in any way the title from the book, because quantitative reasoning refers to only a part of mathematics, not to all of it.
I belive this is what was attemped in France (they taught set teory in an abstract way in elementary school), but with poor results and abandoned this approach after some time.
I read the sample chapter: it's surprisingly bad! (He's a good writer normally.)
It's boring, and badly-written (the Greeks didn't use symbols like x, instead they used x), with poorly-chosen examples (curtains not a good example of abstraction; correlated risks a good example of correlation, which is off-topic here, but a bad example for algebra), and lacking pictures that would help (the wine barrel).
Nothing so esoteric — it’s written for the popular audience. But there’s a lot of “what is mathematics?” or “how do we do mathematics?” Some random highlights:
> A big part of mathematical training is becoming acquainted with different concepts and recognizing how they’re cousins to each other.
> I encourage everyone to fool around with numbers—some enthusiasts discover important patterns that way, and some famous mathematicians, like the late John Conway, made a career of it.
> When we give word problems in school, we tend to cherry-pick perfect models that have one correct answer. Rather than teach a preordained procedure, we could encourage exploration by asking students to fix more realistic models where things are slightly noisy or slightly incorrect.
Perplexing how it's on the front page then given the background of the median HN user (presumably technical with some STEM background in general, some pure VC bros peppered in etc).
Eh, I just like reading new books on mathematics, even if they're covering well trod ground. I think a lot of people in math are like this. I'm not endlessly grinding through introductory textbooks, but there is something nice about seeing someone brilliant give their take on something fundamental. And the cultural differences you get in reading different texts from different periods or from different authors is really stimulating, for me at least.
Basic knowledge resources are very common in HN in front page. And such resource from someone known to write very well will be helpful for newbies even from lesser known personalities. Shared here all the time so why pretend its not common.
I'm not talking to another member or referring to a particular comment, I'm making a brief comment by analogy regarding a pair of public figures who profit from their notoriety.
Moreover I was accurate, unless your idea of kindness is to imply that comparison to a success physicist is "unkind".
That math isn't being discussed here, it's barely even discussed in the book! Frankly how often do you open an HN thread on the front page and see any significant technical discussion of anything other than coding? I find the times I see someone writing out equations and discussing fundamentals so rare and welcome as to be memorable.
Most conversation here is profoundly non-technical and never touches anything like "the math".
I pre-ordered some time ago, and received a copy this morning. I love it.
It's direct and well-written; a high-level overview of mathematics from numbers to analysis and dynamics. I think it's well organized, if brief at ~100pp.
This would be a great read for laypeople as well as a potential gap-filler for anyone in STEM.
I haven't read Tao's book yet, but Lang's very ... schooly kinda book. From what I gather of the Tao's ToC it's more approachable as a non-fiction type of book.
The "algebra excerpt" [1] doesn't impress me at all. Sure, maybe equations intimidate some, but why not add some images or figures? Is that page really one of the six chapters, or is it 1/10 of a chapter?
Tao's essentials are, "... numbers as the gateway to quantitative thinking; algebra as the gateway to abstraction; geometry as a way to calculate beyond what we can see; probability as a tool to navigate uncertainty with rigorous thinking; analysis as a means to tame the very large or the very small; and dynamics as the mathematics of change."*
Does Tao intend that as the 100,000 ft view of all mathematics, a way to categorize (almost) all of it, the roots of the mathematics tree? Are these the essentials or just some valuable ones.
For comparison, Gowers begins a much larger book with, "there is a crude division that undoubtedly works well as a first approximation, namely the division of mathematics into algebra, geometry, and analysis." [0]
There must be other schemes out there? What are the respected ones?
[0] Timothy Gowers, ed. The Princeton Companion to Mathematics. p.1 (2008)
> There must be other schemes out there? What are the respected ones?
I don't think there are any other divisions there, not really. Disputing such soft statements on algebra and geometry would be like claiming that the horror genre is all about making people laugh and comedy movies are produced to induce fear.
Gowers' division misses probability, though, which really is its own thing. Similarly, counting is the foundation for the axiomatization of mathematics and probably deserves its own category. They all bleed into one another (analysis uses counting for its formalization, probability uses both counting and analysis), but they can each stand on their own informally (and did, in some cases for centuries).
I got this book today, as I had it on pre-order. It's a bit disappointing because, on a quick count, there are only five pages with pictures. And I think I saw just a few equations. It's a bit strange to leave out two essential mechanisms for how mathematicians work and think in a book named Six Math Essentials.
Terence Tao explains 6 essential mathematical concepts: https://www.youtube.com/watch?v=OOMx2BHHWtE
[1] https://www.amazon.com/dp/0374621799?tag=macmillan-20
1. numbers as the gateway to quantitative thinking
2. algebra as the gateway to abstraction
3. geometry as a way to calculate beyond what we can see
4. probability as a tool to navigate uncertainty with rigorous thinking
5. analysis as a means to tame the very large or the very small
6. dynamics as the mathematics of change
That's a historical artifcat, not an innate property. (Math was first invented in the Bronze Age for accounting and architecture.)
There's a theoretical parallel universe with a mathematics where numbers aren't first on the ladder but down there around number five. (Ironically computers are an example of that. Figuring out how to do numbers when all you have is finite sequences from an alphabet is advanced CS.)
Secondly there are lots of mathematical objects that significantly predate the bronze age (eg https://en.wikipedia.org/wiki/Ishango_bone but there are others) and because these objects are so old we have no idea why math was first invented.
Thirdly I have no idea the relevance of the theoretical “late numbers “ parallel universe but this thread is about a book in this one.
This isn't true though. It's not supported by anything except Bronze Age tradition.
I.e., when teaching kids advanced math at an early age it's often better to skip numbers altogether and start with set theory or logic. Many kids find that a much easier gateway.
By definition, quantities, either discrete or continuous, as already classified by Aristotle, are the properties that can be expressed by numeric values, as a result of counting for discrete quantities, or as a result of measuring for continuous quantities.
Properties that are not quantities, but which can be compared, to determine which is greater or less, or if they are equal, are qualities.
The simplest numbers, i.e. cardinal numbers a.k.a. natural numbers, are equivalence classes of sets, so indeed the theory of numbers is built upon that of sets (a.k.a. multitudes in older language), but this does not contradict in any way the title from the book, because quantitative reasoning refers to only a part of mathematics, not to all of it.
What do you mean by "often better". Has it even ever been tried out?
Of course, and it's standard practice in some places.
P.S. I'm talking about extracurricular math clubs/circles. School curriculum is a separate thing that's divorced from actual math.
https://www.amazon.co.uk/Six-Maths-Essentials-Model-World/dp...
That said, you can switch to the Kindle version and request a sample. The kindle samples usually have a ToC.
Or click on the link to a sample chapter, listed on the submission:
https://www.sciencefriday.com/articles/six-math-essentials-b...
It's boring, and badly-written (the Greeks didn't use symbols like x, instead they used x), with poorly-chosen examples (curtains not a good example of abstraction; correlated risks a good example of correlation, which is off-topic here, but a bad example for algebra), and lacking pictures that would help (the wine barrel).
> A big part of mathematical training is becoming acquainted with different concepts and recognizing how they’re cousins to each other.
> I encourage everyone to fool around with numbers—some enthusiasts discover important patterns that way, and some famous mathematicians, like the late John Conway, made a career of it.
> When we give word problems in school, we tend to cherry-pick perfect models that have one correct answer. Rather than teach a preordained procedure, we could encourage exploration by asking students to fix more realistic models where things are slightly noisy or slightly incorrect.
Perplexing how it's on the front page then given the background of the median HN user (presumably technical with some STEM background in general, some pure VC bros peppered in etc).
The Tao love conquers all, as it were.
> Be kind. Don't be snarky.
Moreover I was accurate, unless your idea of kindness is to imply that comparison to a success physicist is "unkind".
Most conversation here is profoundly non-technical and never touches anything like "the math".
It's direct and well-written; a high-level overview of mathematics from numbers to analysis and dynamics. I think it's well organized, if brief at ~100pp.
This would be a great read for laypeople as well as a potential gap-filler for anyone in STEM.
Is this a book for children?
[1] https://www.sciencefriday.com/articles/six-math-essentials-b...
Does Tao intend that as the 100,000 ft view of all mathematics, a way to categorize (almost) all of it, the roots of the mathematics tree? Are these the essentials or just some valuable ones.
For comparison, Gowers begins a much larger book with, "there is a crude division that undoubtedly works well as a first approximation, namely the division of mathematics into algebra, geometry, and analysis." [0]
There must be other schemes out there? What are the respected ones?
[0] Timothy Gowers, ed. The Princeton Companion to Mathematics. p.1 (2008)
I don't think there are any other divisions there, not really. Disputing such soft statements on algebra and geometry would be like claiming that the horror genre is all about making people laugh and comedy movies are produced to induce fear.
TIL about Quanta Books, thanks OP — will be checking this out! :D