The Limits of Mathematical Equality

Listen to Kate read: The Limits of Mathematical Equality

Where we learn that it’s all in the relationship.

The whole of this site is finding the heart of math in order to navigate real life. The whole of this essay is understanding how equality is both defined and perceived in math to see if it could offer some guidance toward dismantling the limitations that define equality – and inequality – in our everyday world.

I chose an easy lift for this essay, right? 😅

Here’s what I started with: when two things are considered to be quantitatively exactly the same, they are said to be equal. The equals sign ( = ) is a mathematical symbol that is used to show equality in a distinct sense. Ok, great, got it.

Then I went down one of those sorta daisy-chained research holes. Wolfram (the math resource, not the person himself) used the phrase ‘well-defined sense’ (meaning: unambiguous) in its definition of the word. So according to them, for two things to be equal they must be unambiguously — like, no questions remaining to the contrary — quantitatively the same. And my always trusty Harper Collins Math Dictionary says that equality is “a statement indicating that the quantities on either side of the equals sign ( = ) are equal in value.”

Ok, so, since we’re not all carrying around an equals sign with us to gather ’round, I decided to put on my etymological glasses and consider that since the suffix ‘-ity’ added to a word simply means we’re talking about the ‘state of being’ of said word, then equality would mean the state of being quantitatively the same. Across the board. As in, having the same value.

Well, who gets to decide the value of something? Or someone.
What would – or could – that even mean?

Mathematics and money both originated in abstraction… Substituting numbers for objects changed the world, for better or worse. At once, everything became quantifiable…

“Thinking in Numbers: How Math Illuminates Our Lives” by Daniel Tammet
(pg. 167- 168)

Ah, yes, examining the actual worth of something, a formula that to some degree includes not just quantifying but actually assigning a value to something. And though we’d like to believe that those who get to assign a value are assigning it based on some sort of factual information, so many times – for so so long – that value is assigned subjectively.

Since I apparently just took a cliff dive into equality IRL, and I don’t want to be there quite yet, I’m gonna rewind and explore how the word exists and is used in another field of study. What way does equality show up in music?

It certainly shows up when harmony is performed, and I found that it’s also used in a lil’ something known as musical temperament. Perhaps more commonly referred to as pitch modification, the act of tempering in music means you are altering the physicality of whatever is producing a particular tone. That alteration can be on the instrument itself: on the string that’s reverberating or on a pipe that air is being forced through.

One thing that musical tempering shan’t do, though, is modify that tone into a whole other key. In order to accomplish that, a different sized string or pipe would need to be added onto the instrument, and further tempering could then be applied. But there’s a practical limit to all that – and what if different types of instruments were to be played together, at the same time? It seems like there should be some standardization between the tones and pitches of those instruments so that, you know, music that sounds pleasing to the ear could be collaboratively produced?

Enter: that Greek philosopher and mathematician from more than 2,500 years ago, Pythagoras, who – by mathematically determining what the ratio was for ‘perfectly sounding’ tones within (what would be known as) an octave (that’d be 2:1) as well as the ratio for a ‘perfect fifth’ (that’d be 3:2) – he created a tuning system, one that was based on splitting an octave into 12 intervals.

The thing is – this tuning system of Pythagoras that used two ratios worked out all right for the progression of an octave or two. But because those ratios weren’t fractionally the same, as the number of octaves increased – one after the other – this whole tuning system got more and more out of wack.

Here’s the bit for you to remember in this whole music reference: This tuning system is known as equal temperament. And it is the prevalent tuning system that is still used today.

Oh look, a system built a zillion years ago that is still used today.
That always works out for the best.

The primary feature of equal temperament is uniformity.

But equal temperament, much like the very concept of normality, is a product of its time and place. It reflects the politics of a culture that values consistency over variety, uniformity over difference.

“Temperamental Differences” by Blake Howe (Professor of Musicology)
from the blog, The Avid Listener

Sure, Bach and Mozart and other noted musicians through the years created other forms of tuning – known as well temperaments – where the key feature of the tuning is variety, not uniformity. So, yes, octaves are tuned based on their 12 intervals, but then based on the music at hand that’s being played, though the tuning of the most used ‘perfect fifths’ would draw guidance from Pythagoras’ ratios, it wouldn’t be at the expense of messing up the tuning of the octaves of the instrument(s), as what happens when solely equal temperament would be used.

Hrrrm. This is the point in my research where I decide to come at my topic from a different angle. All I’m getting here is that the word equality – and its root word, equal – are about subjectively produced quantitative values that are used to promote uniformity as the norm. This information is deflating.

I decided to look at the word inequality and almost immediately I skidded into this sentence (from a site producing content for home-schooling, no less):

Inequalities do not represent an exact amount
but instead represent a limit of what is allowed or possible.

Equations and Inequalities: Real_World Situations” from Elephango

🧐 You know, that sounds about right. 🤨 If you (or me, or that person over there) is not able to be described as being a part of the Equal Gang, then you (and I, most certainly) represent a limit of what is allowed or possible. Shit, I’ve been there. More times than I can count. That all sounds about right (‘right’ as in ‘familiar’, as opposed to ‘correct’).

Math’s history with equality and inequality is, surprise surprise, deeply integrated into the formulas behind everything from the models we base insurance rates on to the algorithms that rule inform our lives today in the 21st century.

Poor people are more likely to have bad credit and live in high-crime neighborhoods, surrounded by other poor people. Once the dark universe of WMDs (weapons of math destruction) digests that data, it showers them with predatory ads for subprime loans or for-profit schools. It sends more police to arrest them, and when they’re convicted it sentences them to longer terms.

“Weapons of Math Destruction: How Big Data Increases Inequality
and Threatens Democracy”
by Cathy O’Neil
(pgs. 199 – 200)

What to do about the usage of the word equal and all its variations IRL? I do find some personal satisfaction in the following suggestion from Simone Weil, 100 or so years ago. She proposed using mathematical proportion to integrate equality with differentiation. So where the concept of ‘equal’ in the equal temperament tuning system was all about a consistent uniformity, this is proposing creating an equality in the midst of what are already existing differences.

For instance, an employer who is incapable or guilty of an offence against his workmen ought to be made to suffer far more, both in the spirit and in the flesh, than a workman who is incapable or guilty of an offence against his employer. Furthermore, all workmen ought to know that this is so.

It would imply… a conception of punishment in which social rank, as an aggravating circumstance, would necessarily play an important part in deciding what the penalty would be.

“Simone Weil: An Anthology” edited by Siân Miles
(pg. 100)

I mean, I get the complications that could arise with this methodology 🥸 But, you know, the higher you are, the harder you fall, blah blah blah, how about using that line of thinking to bring some equality to the justice system… that’s worthy of a chat. 😉

Jumping out of this philosophical math route and getting back on track with those mathematical factoids, I am – as are you now, too – rewarded. Let me share with you the birth of Category Theory and its birthers, the mathematicians Samuel Eilenberg and Saunders Mac Lane. It was 1945 and they wrote a paper titled, General Theory of Natural Equivalences, emphasizing that we should be focusing on what the meaning is of the relationship between things – as opposed to the things themselves – and about categorizing these sets as “collections of a higher order.” Think about it: the way categories are put into order is by the description of how those objects are related to each other – including the ways they may be equivalent to each other, without a care that they are not same-same equal to each other.

The words equal and equivalent are not synonyms, certainly not in a mathematical sense.
When two things are equal, they are the same in all ways. When two things are equivalent, they have a relationship to each other that is similar under certain conditions.

What equivalence allows for that equality doesn’t is relational; it acknowledges the reality of the conditions that objects exist in can change, just as perspectives can change, which can then result in a change in how those objects, or things, (or places? or people?) are quantified and valued.

…there is a growing community of mathematicians who regard the equal sign as math’s original error. They see it as a veneer that hides important complexities in the way quantities are related — complexities that could unlock solutions to an enormous number of problems. They want to reformulate mathematics in the looser language of equivalence.

With Category Theory, Mathematics Escapes from Equality
by Kevin Hartnett for Quanta Magazine 

Hullo to you “growing community of mathematicians,”
where have you been hiding and what’s the password?

One of the leading mathematicians working in Higher Category Theory today is Emily Riehl (this person is fantastic, use the googs to learn more about Emily now – well, actually, wait to do that till after you finish this essay. Please and Thank You).

“Theorems proved 2,000 years ago are still true. But perspectives change,” Riehl says. “The notion of equality in mathematics is way too restrictive. It’s just very uncommon that you’re going to find things that are actually equal, even when things are, in some very strong sense, the same. Category theory gives us a rigorous language for expressing the sameness between things that aren’t literally equal.

“The Mathematical Mind of Emily Riehl”
by Dale Keiger
John Hopkins Magazine, Winter 2021

I want to create a new habit for myself – swapping out the word ‘equivalence’ for ‘equality’ wherever I can. Acknowledging that the word ‘equality,’ not just mathematically but IRL, doesn’t and hasn’t allowed for the conditional changes that naturally occur over time to be taken into consideration on whether something, or someone, is equal or unequal.

Despite everything written here today, I still believe and therefore will still end this essay with my same ending phrase, because one thing I know for sure is that it’s the humans behind the math – the ones developing those algorithms now just as their ancestors created formulas in millennia past – that insert bias into their work. Math still has our back, I am sure of it.

Thanks math, you’re the best.


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  • Have you been reading my Kindle wish list recently? 🙂 A few books on category theory, including one by Emily Riehl, and a few books on alternate musical tunings! I haven’t combined them yet, however.

    • Ha! I should be asking you – have you been reading my research notes? 😁 I really like that you’ve (perhaps unconsciously) connected those worlds, too!

      • I pretty much spend all of my free time (which is when I’m awake) on microtonal computer music / music theory these days. For some reason category theory popped up a few days ago and I don’t quite know why Kindle’s recommender did that.

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