The Grammar of Mathematics

Aug 23, 2021 | Location West County

The Grammar of Mathematics

Cole
Mar 28 · 4 min read
 
 

“The universe cannot be read until we have learnt the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word.” -Galileo Galilei

I’ve always loved this quote. In it, Galileo lays out his philosophy very simply: math is the language of the universe. This was a novel idea for the time; most looked to the church to explain natural phenomena. Galileo used this mindset to make massive strides in our understanding of physics and astronomy, completely reshaping our view of the universe.

The reason this quote has stuck with me is the analogy it creates between math and language. Languages are complex, ever-evolving forms of communication that allow us to convey ideas. Mathematics aims to do something similar, it’s just specialized for a set of particularly abstract concepts. Due to this need, mathematical notation must be unambiguous (more on this later). This has led to an immense amount of mathematical symbols, created to distinguish between an ever-growing list of ideas. Many of these symbols are combined with methods such as subscripts and superscripts to further distinguish between objects within mathematics.

Like language, mathematics contains a variety of dialects. Different fields of mathematics have different needs, and the notations within these fields evolve to reflect them. This is a long, difficult process. Newer fields, like category theory, are currently working to create their own “standards.” Most importantly, all aspects of math have a shared commonality in their notation, keeping them united under one language.

How precise?

Earlier, I said that mathematical notation must be precise in order to accurately convey an idea. This is much easier said than done. Because math is ultimately a human endeavor, with thousands of us working to expand our sphere of knowledge, this cannot be as rigorous as we’d like. See this page for several examples of differences in notation that can potentially lead to confusion. When I was studying math and physics for my undergraduate degree, I took Multivariable Mathematics and Mathematical Physics in the same semester. Both classes dealt with Stoke’s Theorem, but used radically different notation and wording to describe it. Balancing these two similar versions of the same theory was a huge challenge for me. I can only imagine the pain of reading through mathematical papers, each with its own take on how symbols should be used.

Of course, the constant shift in mathematical notation is essential. Without it we would still be stuck with Roman Numerals which become incredibly clunky for large numbers. Richard Feynman famously invented his own notation for sine and cosine which, while pretty, never caught on (shown below).

My absolute favorite article on Stack Exchange is linked here. It contains a bunch of alternate notation for logarithms, exponents, and roots: three concepts that are intimately related. However, their respective notations don’t indicate this. Commenters come up with several beautiful ideas; it’s a fascinating read. Looking through these suggestions will help you understand the ideas I present in the remainder of this article. If you want a video summary, 3Blue1Brown has a fantastic presentation on it.

Is precision the only goal?

The discussion linked above leads to an interesting question: should we change the notations for logarithms, exponents, and roots? Our current symbols are certainly precise; they convey exactly what we need to know to perform the operation. They still somehow feel like they’re missing something. I strongly sympathize with the question-asker in that Stack Exchange discussion.

I distinctly remeber being baffled by logarithms when they were first presented to me. I only made it through high-school math by blindly memorizing what it does and how to calculate the logarithms provided to me. Only much later in my mathematical education did I realize the strong relationship that connected logarithms to exponents and roots. Of course, this is only a personal anecdote, but it is interesting to consider how this could be avoided. Would this lapse not have occured if I had been presented with a different notation?

Believe it or not, the symbol π is also controversial. There is a group dedicated to replacing π with τ (pronounced tau), where τ = 2 * π. At first, this seems rather silly. Why is this necessary? There is a lot of culture around π, most notably π-day which occurs on March 14. In addition, an immense amount of mathematical textbooks would have to be rewritten. Proponents of τ have written out an extensive list of reasons for this change. While they are presented in a joking manner, the arguments laid out are rather convincing, and I would encourage you to read it. It may improve mathematical education when τ is used instead of π, but this is yet to be proven. (I happen to be a supporter of τ only because I was born on τ-day).

Communicating mathematics is incredibly important. Just like a language, math is all about expressing complicated ideas in simple ways. It’s important to continue discussing how we talk about math and how it could be made better. If you’re interested in notation and how it can be used to describe more ideas (not just for mathematics) then I highly recommend you check out this collection of pieces. It served as my inspiration for this article (some of the pieces are linked already) and has so many cool thoughts in it!

 

 

 

 

 

 

 

Locations near
ME
Ashburn 0.93 mi
43330 Junction Plaza
#160
Ashburn, VA 20147
Sterling 4.07 mi
44 Pidgeon Hill Drive
#100
Sterling, VA 20165
Leesburg 5.69 mi
521 E Market St
#B
Leesburg, VA 20176
Herndon 7.66 mi
2465 Centreville Road
#J2
Herndon, VA 20171
Stone Ridge 7.88 mi
42020 Village Center Plaza
#100
Stone Ridge, VA 20105
Reston 8.62 mi
1424 North Point Village Center
Reston, VA 20194
Purcellville 12.72 mi
1020 E Main St
Ste L
Purcellville, VA 20132
North Potomac 13.58 mi
12150 Darnestown Rd
Gaithersburg, MD 20878
Centreville 14.52 mi
5959 Centreville Crest Ln
Centreville, VA 20121
Fairfax 14.67 mi
11891 Grand Commons Ave
Fairfax, VA 22030
Potomac 14.98 mi
10232 River Road
#B
Potomac, MD 20854
Germantown MD 15.03 mi
12800 Middlebrook Road
Germantown, MD 20874
Tysons 15.45 mi
328 Maple Ave E
#A
Vienna, VA 22180
Haymarket 17.87 mi
15125 Washington St
Haymarket, VA 20169
Rockville 18.19 mi
20 Courthouse Square
#106
Rockville, MD 20850
Mclean 18.23 mi
1320 Old Chain Bridge Road
Suite #190
McLean, VA 22101
Gaithersburg 18.71 mi
9132 Rothbury Drive
Gaithersburg, MD 20886
North Bethesda 20.42 mi
5268 Nicholson Ln
#N
Kensington, MD 20895
Falls Church 20.59 mi
6674 Arlington Blvd
Falls Church, VA 22042
Manassas 20.65 mi
9722 Liberia Ave
Manassas, VA 20110
Bristow 20.71 mi
12705 Braemar Village Pz
Bristow, VA 20136
Burke 20.82 mi
9411 Old Burke Lake Rd
#C
Burke, VA 22015
Bethesda 21.3 mi
4918 Fairmont Ave
Bethesda, MD 20814
Annandale 21.64 mi
7000 Columbia Pike
Annandale, VA 22003
Damascus 22.77 mi
9815 Main St
Damascus, MD 20872
Arlington, VA 23.29 mi
4801 1st St N
Arlington, VA 22203
Cathedral Heights 23.51 mi
3706 Macomb St NW
Washington, D.C., DC 20016
Olney 23.96 mi
18157 Village Center Dr
Olney, MD 20832
Downtown Silver Spring 24.94 mi
1133 East West Highway
Silver Spring, MD 20910
Lorton 25.06 mi
9027 Silverbrook Rd
#A
Fairfax Station, VA 22039
West Washington 25.69 mi

Washington, DC 20009
Alexandria City 25.7 mi
4605 Duke Street
Alexandria, VA 22304
Lake Ridge 26.5 mi
12473 Dillingham Square
Lake Ridge, VA 22192
Frederick North 26.7 mi
905 W. 7th Street
Frederick, MD 21701
Northern Silver Spring 26.75 mi
732 Cloverly Street
Silver Spring, MD 20905
Alexandria 27.17 mi
6483 Old Beulah Street
Alexandria, VA 22315
Warrenton 27.42 mi
512 Fletcher Drive
Warrenton, VA 20186
Mount Airy 28.24 mi
411 E Ridgeville Blvd
Mt Airy, MD 21771
Capitol Hill DC 28.48 mi
621 Pennsylvania Ave SE
1st-floor unit
Washington, DC 20003
Dale City 28.99 mi
5512 Staples Mill Plaza
Dale City, VA 22193
Mount Vernon 29.66 mi
7696 H Richmond Hwy
Alexandria, VA 22306
Beltsville 30.89 mi
10914 Baltimore Ave
#B
Beltsville, MD 20705
Clarksville 31.56 mi
12250 Clarksville Pike
#D
Clarksville, MD 21029
Laurel 35.02 mi
10095 Washington Blvd N
#136
Laurel, MD 20723
Woodmore 35.27 mi
9101 Woodmore Centre Drive
Lanham, MD 20706
Glenn Dale 35.83 mi
10559 Greenbelt Rd
Lanham, MD 20706
Sykesville 36.57 mi
1207 Liberty Rd.
#D-104
Sykesville, MD 21784
Ellicott City 37.11 mi
3290 Pine Orchard Lane
#B
Ellicott City, MD 21042
Winchester 37.62 mi
2512 S Pleasant Valley Rd
Winchester, VA 22601
Columbia 38.19 mi
8827 Centre Park Drive
#F
Columbia, MD 21045
Stafford 39.58 mi
263 Garrisonville Road
#104
Stafford, VA 22554
Bowie 41.26 mi
15231 Hall Rd
Bowie, MD 20721
Waldorf 42.24 mi
3022 Festival Way
Waldorf, MD 20601
Crofton 42.6 mi
1153 Route 3 North
#120
Crofton, MD 21054
Owings Mills 44.15 mi
9433 Common Brook Rd #100
Owings Mills, MD 21117
Pikesville 46.84 mi
1433 Reisterstown Rd
Pikesville, MD 21208
Dunkirk 49.92 mi
10735 Town Center Blvd
#7
Dunkirk, MD 20754