About a tyrant called "=".

Cornwell1

I got involved in an interesting thread on symmetry and read the following, in my eyes very wise words:

In mathematics, it is often said that the left hand side of the equation represents the very same thing as the right hand side, a specific mathematical value, or object. In reality, the two sides express two distinct things, with an equality between these two. When two things which are different, are said to be equal, the difference between them has already been excused in that judgement of equal. So we now have a second level of excusing differences for the sake of symmetry, the excuse which exists right at the level of producing the equation.

The symmetry of what's the left hand side (LHS) and right hand side (RHS) of the "is equal to" sign, =, is here discussed. It's pretty obvious that = is justified if both sides are the same.
Still, when looking at mathematical equations, there are quite different things on both sides of the = sign. Only trivial cases like x=x look symmetrical. Most equations are not reducible to the trivial example.
So in what sense are both sides equal? Is it only an equality of a quantity or a number?

You can write a condition on energies, say that the kinetic energy equals potential energy. The quantities are the same on both sides, Joule, that is. Dimensional analysis is, by the way, a useful tool if both sides of a = sign are consistent. In the equation of two energies, this is obvious but in complicated expressions it comes in handy and you can even use it to anticipate.

So equality in units is important as well as the numerical equality. What else can = be applied to? It can be used to equalize sets, for example. Or to define a function, f(x)=y. The sign has the power to equalize. It equalizes two expressions on both sides of it. There are variations: <, >, ≤, ≦, ⩽, ≥, ≧, ⩾, and ≠ are cousins of the almighty equalizer but less powerful in the sense that they allow more numericals. Still, it requires both sides to be numerical or sets. Or don't they?
There is another set of nephews: ≈, ∼, ≃, ≅, ≡ each comparing the two sides of an equation and meaning that they are similar or about similar and about equal. ≡ means "identical to" but is not often seen. There is :=, which means equal by definition, and last but not least, my favorite, ≠, not equal to. A bastard child that can be applied to most things in the world.

Is math powerless without =? Is the = the tyrant who equalizes both sides to his advantage, and if so, what's the advantage? Or is he unifying different sides and offering understanding and peace? Is unifying the same as equalizing? If both sides are equal, are they unified? Enough questions. Whatever thoughts related to this thread are welcome!

universeness

↪Cornwell1

I think the concept missing from your OP is balance. Equal means both sides are balanced.

in Computing := means 'becomes equal to.'
So a program variable is given an initial value using :=
For example, using camel casing:

personName:= Jinty

The textual variable personName becomes equal to Jinty
OR 'is instantiated' to Jinty

= is more about balance than symmetry.
A tonne of feathers is not symmetrical in size to a tonne of steel but they are both balanced in weight.

Cornwell1

↪universeness

:ok:

I hadn't looked at it that way yet. I had an equality of masses in mind, but your example shows what it actually is. Literally a balance! The numerical values are equal, but the stuff on one side is different from the other indeed. Balanced but not equal. M(feathers)=M(steel). Equality=Balance...

tim wood

And it's useful to think of these things not so much as what they are, because in a sense they are not anything at all, but instead what they're for, what they're used for, and how they're understood in use.

universeness

↪Cornwell1

:up: :up: :up:

Cornwell1

↪tim wood

To question them is what philosophers do.

universeness

↪Cornwell1

↪tim wood

I think it is the balance that matters most, yes, rather that whats on either side of =

Balance....imbalance.....return to balance
chaos...combination....order....entropy..chaos...combination....order.......

waveforms have balance points or tipping points, in-between a single crest and trough, so you have:
balance.....up crest.....tipping point(or perhaps another balance point)....down crest.....balance.....down trough.....tipping point......up trough.....balance. There are many symmetries in waveforms as well.
The concepts of Balance, Imbalance, Chaos, Order are certainly very important in Philosophy, I think.

Cornwell1

↪universeness

Indeed!

Balance

andrewk

The OP relates to a point that is important in mathematics, yet rarely taught.

In short: an equation, by which I mean two sets of symbols connected by an '=' sign, constitutes only part of a complete logical sentence, not a complete sentence in itself.

The equation needs to be included in a sentence in order to be meaningful. Consider the following:

(1) sin x = tan x cos x
(2) sin x = 0.5

Item (1) is an 'identity', and is true for all x except odd multiples of 90 degrees. So we need a sentence to give it meaning, such as:

"For all real x except odd multiples of pi/2 we have sin x = tan x cos x"

Identities like this allow us to simplify and sometimes solve algebraic problems by substituting the expression on one side for the expression on the other. They do not say the expressions are the same. They are not, as they use different symbols and require different steps to evaluate them. But the identity says that, subject to any constraints imposed by the enclosing sentence, both sides will deliver the same value when evaluated, regardless of the values given to any pronumerals ('variables').

Things like Item (2) are typically just referred to in practice as an 'equation' (a more constrained meaning than I gave that term above) and is used to find allowable values of x. An example might be:

"Indiana Jones worked out that the sun would shine onto the crypt only when the angle of the sun from the vertical was 60 degrees, where sin x = 0.5."

Note that here the equation does NOT say that sin x is always equal to 0.5 but rather, that the thing we are interested in (illumination of the crypt) occurs when sin x has that value.

Conclusion: we can't identify a meaning for '=' without further context, including at least the full sentence in which the 'equation' occurs. That meaning will vary depending on the context.

universeness

↪andrewk

Your conclusion is correct and it would be more helpful to write your second equation as a question, or a statement to be actioned, such as:

(Find values of x such that) sin x = 0.5 (and both sides of this system will be balanced)

To me, that's what sin x = 0.5 means in my head. sin x = 0.5 is just 'shorthand' is it not?

kudos

↪Cornwell1

Words and mathematical symbols often form separate spheres of meaning despite their basis on comparable activities of the mind. Symbols in mathematics are used less frequently as a social language than their alphabetic counterparts. We tend to think of verbal languages as fulfilling social needs and mathematical symbols as fulfilling contingent personal needs, because writing down symbols in mathematics can help us to conceptualize certain kinds of relationships more effectively than verbal ones can. Those symbols tend to take on a life of their own, eventually through habitual usage becoming fixed in their personal-narrative meaning.

At the heart of it, it sounds like the question is about the absolute linguistic unit versus the relational. We find that 'equals' has a meaning in the mathematical community that is more intentionally defined than the one in our casual usage. Do we hold fast to those meanings and furtively ignore their linguistic significance in favour of reproducing completeness, or accept the synchronic meaning freely as separate and referential to a mathematically absolute definition? I think in any feedback system we have to understand there is an underlying dicrete-ness to its idealization; that – excluding special cases – something can be determined as one thing and simultaneously negate another, not solely in a formal sense but also in terms of content.

When we perceive we can negate as well as posit; create something that cannot have being at the same time we imagine some other aspect of it as 'being there.' In our outer world there is thought to be one set of physical laws that negate others, but there could be other physical laws present not meeting certain conditions of sensibility.

I digress...

jgill

N=1/N has two very different meanings in practice. Context means everything.

Agent Smith

a) 2 = 1 + 1 (is)

b) 6 = 12 ÷ 2 = 2 × 3 (is the same as)

c) Left to the reader as an exercise

d) Ditto
.
.
.

Cuthbert

It's an interesting question. Here's a sum for our early years arithmetic class:

3 + 5 = ?

The expected answer is '8'. But suppose we write:

3 + 5 = 5 + 3

No point, red cross, wrong answer. But why? 3 + 5 = 8 is true, and 3 + 5 = 5 + 3 is also true. 3 + 5 = 5 + 3 tells us something that 3 + 5 = 8 does not, namely, that addition is commutative. So it carries new information, just as 3 + 5 = 8 carries new information. We could also write "3 + 5 = x + 3 + 5 - x, for any number x", showing that there is an infinity of solutions to the problem. This will either get us sent home early or promoted to the fast-lane maths class, depending upon our teacher's mood.

Perhaps the tyranny is not the "=" sign itself but the baggage of expectations we carry around with regard to its application.

universeness

N=1/N has two very different meanings in practice. Context means everything — jgill

From reading other comments, where you gave some info about your background, your Maths is far beyond mine. I noticed that N=1/N seems to only have the solution N=1, is this the only solution?

Metaphysician Undercover

↪universeness

What about -1?

You can write a condition on energies, say that the kinetic energy equals potential energy. The quantities are the same on both sides, Joule, that is. Dimensional analysis is, by the way, a useful tool if both sides of a = sign are consistent. In the equation of two energies, this is obvious but in complicated expressions it comes in handy and you can even use it to anticipate. — Cornwell1

The thing about equating kinetic energy with potential energy is that it seems to involve some kind of category mistake to describe the two as equal. One is a measure of the actual movement of a thing, while the other is a measure of a thing's capacity to move. Since a cause is required to transform the potential to actual, then if we express the two as equal we neglect the reality that one is temporally prior, and the other posterior. This temporal difference implies that the supposed equality between them neglects an important fact.

Cornwell1

↪universeness

I wondered about that too. Seems only 1 and -1 seem to be the answer. Maybe N can be different things. So that it's inverse is equal to itself. An inverse matrix?

universeness

↪Metaphysician Undercover

Yeah, another doh! moment for me, to add to my ever-growing collection.
Thanks for correcting the oversight. Same to

↪Cornwell1

So apart from Crowell1's inverse matrix question, it seems to me that all
N=1/N suggests is positive = positive and negative = negative

I think if you write positive = negative, you do require supporting context such as
Balanced/equal in quantity but opposite in 'charge?', 'Polarity?', 'magnetic attraction?'
Which would be more accurate or are they all equally valid.
I've always considered +ve and -ve numbers to owe their existence to the existence of +ve and -ve charge, and its electro/magnetic components.

Cornwell1

↪Metaphysician Undercover

Indeed. Potential energy is different from kinetic energy. Potential energy is defined in force fields. Kinetic energy is caused by a force. So is potential energy but in a "reverse mode". By pulling or pushing, it is stored or extracted. The kinetic and the potential can be transformed into one another or be compared in value. While kinetic energy is always positive, potential energy can be both, depending on the gauge. Is kinetic energy somehow gauge dependent too?
So, values balance, the "things" are different.

Yeah, another doh! moment for me, to add to my ever-growing collection. — universeness

I had too look up "doh":

DOH
Also found in: Dictionary, Thesaurus, Medical, Financial, Encyclopedia, Wikipedia.

Category filter: Show All (22)Most Common (1)Technology (3)Government & Military (9)Science & Medicine (3)Business (2)Organizations (4)Slang / Jargon (6)

AcronymDefinitionDOHDeliriously Overcome with Hilarity (chat/internet)DOHDepartment of Health (various locations)DOHDirectorate of Health (various locations)DOHDepartment of HealthDOHDukes Of HazzardDOHDepartment of Hydrology (various organizations)DOHDivision of HighwaysDOHDetroit Opera House (Detroit, MI)DOHDepartment of HousingDOHDepending on HeelsDOHDestination Option HeaderDOHDocument Operations HandbookDOHDepartment of Highways (Thailand)DOHDate of HireDOHDeclaration of Helsinki (medical ethics; World Medical Association)DOHDays on Hand (inventory)DOHDoha, Qatar - Doha (Airport Code)DOHDefenders of Honor (gaming, Counter-Strike: Source Clan)DOHDouble Over Head (waves)DOHDepartmental Overhead (USACE)DOHDNS (Domain Name System) over HTTPS (Hypertext Transfer Protocol Secure)DOHDepartment of Hell (gaming)

I think the last one doesn't apply in your usage. But I got the feeling! :smile:

universeness

I think the last one doesn't apply in your usage. But I got the feeling — Cornwell1

Ha Ha.....Now that was an impressive list. :lol:
It is a common sound made by the great philosopher Homer Simpson, from The Simpsons cartoon show.
He says Doh! anytime his brain refuses to work properly.
:lol:

ssu

Is math powerless without =? — Cornwell1

Basically, yes.

Math is all about =.

You see, math has emerged from a need to picture reality around us.

Calculation, addition and substraction has been something that we need for practical uses. Math hasn't developed from some existential philosophical interest, but how to solve practical problems. Even animals can count: if a bird sees three men going to a barn and two of them later come out, the bird can count that one of the men is still inside the barn. And calculation, computation, is all about something is equivalent to another.

Now people might think that < or > would be different, but basically it's applying the same logic. The real opposite to this is the non-computable.

And what does and can mathematics say about the non-computable?

Only that it exists. That there exists mathematical entities that are non-computable. And that's it.

And what do we do when we face a problem that is non-computable?

Well, we surely don't use math to solve it. In fact, we likely don't approach the problem as if there would be one certain true solution for it and that one can deduct it somehow. And to describe it we don't make any mathematical models or formulas, but for example use narrative.

jgill

I noticed that N=1/N seems to only have the solution N=1, is this the only solution? — universeness

Certainly n=1/n has solutions n=1 and n=-1. The other meaning I have in mind is quite different. Hint: I have written hundreds of mathematical programs in BASIC. :cool:

universeness

The other meaning I have in mind is quite different. Hint: I have written hundreds of mathematical programs in BASIC. — jgill

I wrote many programs in my very early days as a teacher in BBC BASIC.
Having to number every code line was fun eh?
Would it not just be

100 INPUT N
110 LET N=1/N
120 PRINT N

So do you mean, that such a program would just display fractions rather than provide values for N which make the equation balance?

A code line containing N=1/N is an assignment expression, not an equation.

Cornwell1

Having to number every code line was fun eh? — universeness

This single line shows already your optimistic outlook on life! Great! :smile:

So by N=1/N you mean the new N becomes the inverse of the old, for example 3 becomes 1/3?

universeness

So by N=1/N you mean the new N becomes the inverse of the old, for example 3 becomes 1/3? — Cornwell1

Yes but you covered this in the OP with the composite symbol :=
Early programming languages did not distinguish between = and := or
Equals(=) and becomes equal to(:=)

In words the code line:
110 LET N=1/N
would be 'Let the numeric variable/container called N become equal to (or contain the answer to), 1 divided by the content of variable N after the line 100 INPUT N has been executed.

BUT this is not the correct mathematical use of =
= is not an ASSIGNMENT operator is is a BALANCE operator
So the equation N=1/N in words is

Find 'anything you like' to put into the algebraic variable N so that both sides of this 'equation' are balanced.

So more modern programming languages use := to act as the assignment operator and the use = as a true/false operator, such as

Condition: IF N=1/N THEN
Action if true: PRINT N
Action if false: PRINT 'inputted value does not balance the equation.'

jgill

I wrote many programs in my very early days as a teacher in BBC BASIC.
Having to number every code line was fun eh? — universeness

N=1/N was a trick question. Sorry :cool:

I now use Liberty Basic. I used Virtual basic until one morning I opened my computer and discovered that Microsoft had deleted the language. Over the years I have tried a number of languages, Pascal, Fortran, Mathematica, C++, etc. But it seems the more sophisticated they are, the more they cater to the popular applications in math. My interests are about as as far from popular as one can be.

I enjoy the challenge of programming a complicated and unusual math process from scratch. Click on my image to see an example.

I haven't had to use required numbering in some time. Neither VB nor LB require them.

Wayfarer

Is math powerless without =? Is the = the tyrant who equalizes both sides to his advantage, and if so, what's the advantage? — Cornwell1

I have the idea that the equals sign represents something uniquely powerful about the ability to reason, and furthermore that it is often taken for granted.

Notice that the expressions "the same as" or "is" are roughly equivalent to "=". In all such cases, whether you're saying that "this thing is the same as that thing" or "28+2=30", you're employing a judgement about identity and/or meaning. In maths there is no room for disagreement about what "=" means, as maths deals only with defined abstractions.

However in cases of practical judgement, where we say that one thing is ‘the same as’ another thing, there may be room for disagreement or interpretation. But the point is, all such judgements rely on rational abstraction. This is something everyone does automatically, as it is intrinsic to the nature of thought and speech. Not only maths, but also logic, wherever the expression ‘is’ is used, exemplifies this capacity. ‘That shade of green is very like the Irish hills’. ‘Income inequality is the cause of social conflict.’ In all cases we’re making judgements about equivalences, which we look through, which is what we bring to bear on any such judgements.

I think this is the source of the idea of ‘mathematical certainty’, which is that only insofar as something can be expressed numerically can it be described with certainty. Mathematical physics, then, achieves its astonishing degree of mathematical rigour and accuracy because the objects of physics are highly amenable to quantisation, their attributes are describable entirely in terms of numerical values in practical terms. I believe that is the source of the prestige of physicalism in contemporary culture; it’s through mathematical physics that an enormous number of astounding scientific discoveries have been made, which has made physics paradigmatic for knowledge in general. It is the ability to reduce and abstract to numerical values that is behind this ability. And that relies at every point on judgements of equivalence.

So - agree that “=“ is all-powerful although would not necessarily concur that this power is tyrannical.

universeness

Click on my image to see an example. — jgill

I assumed by 'my image' you were referring to the icon which takes you to your profile page but when I went there, I could find no code example
Used many programming languages throughout my career as well. Mostly using Python at the moment.

universeness

↪Wayfarer

All sounds good to me!

jgill

I could find no code example — universeness

O0ps, sorry, I meant the icon is a product in the complex plane of a BASIC program I wrote . In this instance, the program created the unexpected demon from a coupled pair of differential equations: dz/dt=f(w,t) and dw/dt=g(z,t),where the functions involved contained sines and cosines.

During the time I worked I knew a number of mathematicians who would have little to do with computers. A bit surprising since numerical analysis was popular then, and some of them were actually researching in that topic. I did too, but computers were primitive and numerical analysis sought to speed up computations, even if the mathematician couldn't speak the CS language.

My initial encounter with computers was a graduate math course in numerical analysis taken in 1962. We wrote short programs, turned them in to someone behind a window where IBM cards would be punched, and finally after a day or so, run through a machine the size of a large room. Then we would find we had made a mistake, and would repeat the process over several days.

It was not a pleasant experience.

Cornwell1

My initial encounter with computers was a graduate math course in numerical analysis taken in 1962. We wrote short programs, turned them in to someone behind a window where IBM cards would be punched, and finally after a day or so, run through a machine the size of a large room. Then we would find we had made a mistake, and would repeat the process over several days.

It was not a pleasant experience. — jgill

And you can do this at home nowadays? Progression can be great!

About a tyrant called "=".

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