# Are 6 + 6 the same as or equivalent to 4 + 8 or 8 + 4 ?

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There are at leat a couple of current threads where a muddle in the question of equivalence is causing difficulty.

I have posted this example for discussion to show that purely mathematical equivalence can be too restrictive.

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55 minutes ago, studiot said:

Are 6 + 6 the same as or equivalent to 4 + 8 or 8 + 4 ?

There are at leat a couple of current threads where a muddle in the question of equivalence is causing difficulty.

I have posted this example for discussion to show that purely mathematical equivalence can be too restrictive.

What do you mean restrictive? This is not always correct, there are numbers which do not comply with this.

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10 minutes ago, koti said:

What do you mean restrictive? This is not always correct, there are numbers which do not comply with this.

I wasn't aware the the integers 4, 6 and 8 were octonions?

No, I was thinking about the circumstances, not the integers themselves.

In any case the threads I referred to were about other things such as relativity models and mathematical limits.

No I was thinking about the common drawing of the conclusion 6 + 6 = 12 therefore 8 + 4, which also = 12, is the same thing in 'my' particular case.

Edited by studiot

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1 hour ago, studiot said:

I have posted this example for discussion to show that purely mathematical equivalence can be too restrictive.

I do not even see a particular example of mathematical equivalence in the question. Unless it gets made more specific.

The symbol "+" is used three times. To make sense of the question it is imperative to assume that it means the same (not "something equivalent") each time. It is not addition of integers in one use, and addition of complex numbers in another, even though we almost always use the same symbol for either.

By default the name + denotes a binary operator, that is, a 2-variable function from a domain into itself, and the default notation is x + y instead of the standard +(x,y).

The question is insufficiently precise to allow us to know what the function + is, in particular its domain. From context we might assume that + means addition of integers and the domain is the set of integers. If this is the intended meaning, then the question becomes equivalent to the question "if f(x) is defined to be 0 for all x, is f(x) the same as or equivalent to f(y)?" In which case the answer is that f(x) is the same, i.e. equal to, f(y), and f(x) is also equivalent to f(y), since equivalence is preserved under identity, by definition.

Can you rephrase the question, so that the meaning becomes clearer, also to me?

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15 minutes ago, taeto said:

Can you rephrase the question, so that the meaning becomes clearer, also to me?

If I sent you down to the shop to buy a dozen oranges and you came back and said that

I could only get 6 (4 or 8) oranges in the first shop so I went to another shop and got 6 (8 or 4) more.

I think you or I could reasonable say that your purchases were all equivalent to the original mission objective.

This is the philosophy section, remember.

But if I asked for nuts and bolts and you came back with:

6 nuts and 6 bolts

4 nuts and 8 bolts

8 nuts and 4 bolts

Would any of these situations be equivalent?

(I think all three outcomes are different)

Edited by studiot

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5 minutes ago, studiot said:

This is the philosophy section, remember.

You typed "mathematical equivalence"  I gather that in "philosophy", + is not a binary operator unless you specify it explicitly

But the elaboration is helpful, so thanks

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8 minutes ago, taeto said:

You typed "mathematical equivalence"

Sorry I meant numerical equivalence.

Thank you for the correction.

You might be suprised at my idea of the difference of the three nuts and bolts example, but I am trying to avoid predjudging the issue in other peoples' eyes.

Edited by studiot

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39 minutes ago, studiot said:

If I sent you down to the shop to buy a dozen oranges and you came back and said that ﻿﻿

I could only get 6 (4 or 8) oranges in the first shop so I went to another shop and got 6 (8 or 4) more.

I think you or I could reasonable say that your purchases were all equivalent to the original mission objective. ﻿

Oranges from second shop could be from different country, different specie, with different parameters (thickness of skin, average weight), than they were in the first shop.

So, I would say, that for cook, it can make difference.

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21 minutes ago, Sensei said:

Oranges from second shop could be from different country, different specie, with different parameters (thickness of skin, average weight), than they were in the first shop.

So, I would say, that for cook, it can make difference.

Yes they could be somewhat different, but they would still be recognisable as oranges.

The nuts and bolts example is fundamentally different.

Here

with 6 nuts and 6 bolts I could make 6 connections with zero waste, enough to bolt together many computer cases

with 4 nuts and 8 bolts I could only make 4 connections leaving 4 wasted bolts.

with 8 nuts and 4 bolts I could again only make 4 connections but this time I could either waste 4 nuts or I could use them as lock nuts making a stronger anti-vibration connection.

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6 minutes ago, studiot said:

with 6 nuts and 6 bolts I could make 6 connections with zero waste, enough to bolt together many computer cases

with 4 nuts and 8 bolts I could only make 4 connections leaving 4 wasted bolts.

with 8 nuts and 4 bolts I could again only make 4 connections but this time I could either waste 4 nuts or I could use them as lock nuts making a stronger anti-vibration connection.

As long as their dimensions and other properties matches..

e.g. 10 mm bolt won't match for 11 mm nut.

Edited by Sensei

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8 minutes ago, Sensei said:

e.g. 10 mm bolt won't match for 11 mm nut.

You are not playing the game. In philosophy every nut matches every bolt

Quickly grabs hat and coat.

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18 minutes ago, Sensei said:

As long aseir dimensions and other properties matches.. th th

e.g. 10 mm bolt won't match for 11 mm nut.

4

10mm bolts are stronger than 4mm bolts.

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2 minutes ago, dimreepr said:

10mm bolts are stronger than 4mm bolts.

Don't they have the same breaking strain or stress?

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1 minute ago, studiot said:

Don't they have the same breaking strain or stress?

per mm yes

27 minutes ago, Sensei said:

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3 minutes ago, dimreepr said:

10mm bolts are stronger than 4mm bolts.

You can't be so sure with 100% certainty.. That depends on their constitution, microscopical anomalies, admixtures, micro cracks. etc. etc. e.g. 4mm metal bolts could be stronger than 10mm plastic bolts.

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2 minutes ago, dimreepr said:

per mm yes

?

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3 minutes ago, Sensei said:

You can't be so sure with 100% certainty.. That depends on their constitution, microscopical anomalies, admixtures, micro cracks. etc. etc. e.g. 4mm metal bolts could be stronger than 10mm plastic bolts.

2

exactly

4 minutes ago, studiot said:

?

44 minutes ago, studiot said:

Yes they could be somewhat different, but they would still be recognisable as oranges.

The nuts and bolts example is fundamentally different.

Here

with 6 nuts and 6 bolts I could make 6 connections with zero waste, enough to bolt together many computer cases

with 4 nuts and 8 bolts I could only make 4 connections leaving 4 wasted bolts.

with 8 nuts and 4 bolts I could again only make 4 connections but this time I could either waste 4 nuts or I could use them as lock nuts making a stronger anti-vibration connection.

3

Im agreeing with you.

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Just now, dimreepr said:

exactly

Im agreeing with you.

Oh.

I thought you were being ambiguous.

:)

They will have the same breaking stress (strain) but different breaking loads.

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But if I asked for nuts and bolts and you came back with:

6 nuts and 6 bolts

4 nuts and 8 bolts

8 nuts and 4 bolts

Would any of these situations be equivalent?

(I think all three outcomes are different)

Mathematics is an artificial and arbitrary imposition of our understanding.   Only one and zero actually exist.

Math "works" because each term is defined and every known operation reflects the logic of nature.

In reality there is no such thing as "6 nuts and 6 bolts" because every bolt and every nut is different than every other.   But in order to use ordinal math it is necessary to treat abstractions such as "6 apples" or "6 bolts" or "6 no units".  This makes even the numbers abstractions, of course.   If you want to count bolts then it is necessary to use our mathematics but it is not the only way to understand apples and oranges.

"Math" is by nature restrictive so it frequently becomes necessary to define terms.   There are an infinite number of pitfalls possible each time we apply math to the real world and these tend to be invisible to us until after the fact.    Generally when communicating mathematical concepts a drawing can prevent most miscommunication.

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Quote

Are 6 + 6 the same as or equivalent to 4 + 8 or 8 + 4 ?

As you say "Are", is the question about the three symbols "6", "+" and "6"? Or did you mean "Is ..." (referring to "6 + 6")?

How do you define "the same as"?

How do you define "equivalent to"?

Are you asking if the character strings are the same/equivalent?

Are you asking if the arithmetic expressions are the same/equivalent?

Are you asking if the results of the arithmetic expressions are the same/equivalent?

Are asking if the meaning of the individual symbols "6", "8", "4" and "+" are the same/equivalent?

I see you have moved on from (possibly) talking about arithmetic to shopping lists, and a completely different question. It is hard to see what the point is.

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so it frequently becomes necessary to define terms.

It is always necessary to define terms in Mathematics.

Sadly I fear the rest of your posting entirely misses the OP which is about Philosophy, not Mathematics.

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"It is always necessary to define terms in Mathematics.

Sadly I fear the rest of your posting entirely misses the OP which is about Philosophy, not Mathematics."

I wasn't referring to defining mathematical terms so much as units.   Math is so restrictive largely because we are trying to impose abstractions (numbers and their operations) on concrete reality.

The nature of math has always been considered a philosophical issue but, like with most things, I tend to disagree.

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"It is always necessary to define terms in Mathematics.

Sadly I fear the rest of your posting entirely misses the OP which is about Philosophy, not Mathematics."

I wasn't referring to defining mathematical terms so much as units.   Math is so restrictive largely because we are trying to impose abstractions (numbers and their operations) on concrete reality.

The nature of math has always been considered a philosophical issue but, like with most things, I tend to disagree.

I have to say I can't see what your posts are adding here.

I also see nothing to complain about so +1 to counterbalance the negative one.

Quote

nothing to complain abou

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7 hours ago, studiot said:

I have posted this example for discussion to show that purely mathematical equivalence can be too restrictive.

I'll try to add a little example where I think physics/engineering gives same result for 6+6 and 8+4:
Moving a certain weight requires force 500N
The weight is moved the distance 1m
Force x Distance 500x1 = 500 J
Doing it for 6+6 times or 8+4 times results in equal amount of Joules.

The above is hopefully correct enough to make the point; mathematics (and also hopefully physical formulas) gives same result.

Now a philosophical case where the math* above is too restrictive;
At the local gym I can do a certain exercise requiring 500N of force 6+6 repetitions within 1 minute if there is a short break between sets of 6. But I can not do the exercise 8+4 repetitions since I'll fail at attempt 7 or 8. In this case the numbers that gives the sum 12 is very important to the outcome; 6+6 is not the same as 8+4. On a good day I think 7+5 may be ok.

*) and also physics, mine or in general?

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8 hours ago, studiot said:

There are at leat a couple of current threads where a muddle in the question of equivalence is causing difficulty.

I have posted this example for discussion to show that purely mathematical equivalence can be too restrictive.

Chronological order may be important.

In culinary art for example, the time you put the ingredients in the recipe is important. A recipe with 6+6 eggs is not the same that a recipe with 8+4 eggs.

The same at war. Dropping 6 bombs today + 6 bombs tomorrow is not the same as dropping 4 + 8.

And there must be more examples. Like stacking objects.

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