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

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I glad to see that some have now caught my drift that just because 6+6 has the same sum as 8+4 it does not mean that 6+6 is the same as 8+4.

Unfortunately some are offering this type of reasoning as a valid argument.

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he example of 6 + 6, 4 + 8 and 8 + 4 is different from 6 bolts and 6 nuts, 4 bolts and 8 nuts and 8 bolts and 4 nuts because the first set are pure numbers, the second are things.  In other words, let x = bolts, y = nuts.  Then your two examples form to

Numerical equivalence:  6 + 6 = 4 + 8 = 8 + 4

But for nuts and bolts 6x + 6y does not necessarily equal 4x + 8y which does not necessarily equal 8x + 4y

Said differently, that which holds with pure numbers need not hold when representing numbers of things.

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7 hours ago, OldChemE said:

he example of 6 + 6, 4 + 8 and 8 + 4 is different from 6 bolts and 6 nuts, 4 bolts and 8 nuts and 8 bolts and 4 nuts because the first set are pure numbers, the second are things.  In other words, let x = bolts, y = nuts.  Then your two examples form to

Numerical equivalence:  6 + 6 = 4 + 8 = 8 + 4

But for nuts and bolts 6x + 6y does not necessarily equal 4x + 8y which does not necessarily equal 8x + 4y

Said differently, that which holds with pure numbers need not hold when representing numbers of things.

Yes you need to separate out the numbers and the things.

Also you could consider more complicated combinations such as the Maximum number of usable components or the Minimum number of wasted components in the bag.

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"I glad to see that some have now caught my drift that just because 6+6 has the same sum as 8+4 it does not mean that 6+6 is the same as 8+4.

Unfortunately some are offering this type of reasoning as a valid argument."

People here almost invariably miss my point so much that they believe I'm not even on topic!

What you're getting toward here is an entirely different way to see math and its reality.   Of course 4 + 8 are never really  equivalent to 6 + 6 in the real world.   There are many many different types, sizes, and compositions of bolts and nuts that have been made in the history of man and they are not always interchangeable with each other and nut and bolts themselves are never interchangeable.  There are an infinite possible types of either nuts or bolts possible with our definitions.   Numbers are necessarily abstractions and as such 4 + 8 = 6 + 6.   But this isn't magic or the nature of reality it's because of definitions.   Rather it is because reality itself is logical and so long as operations on its quantification are logical than these abstractions are numerically logical.

In the real world though we must add 4 apples and 8 apples or 6 apples and 6 apples and we'll always get 12 apples.  But in the real world every set of four ,six, eight, or twelve apples is different.  Every apple in the real world is different so there's no such thing as "two apples".   It is an abstraction.   You can't really put "a" nut on 'a" bolt and must put the nut on the bolt.   In the real world it's not that uncommon for a nut to not fit a bolt but to fit the next "identical" bolt.  There are probably "work arounds" for everything including math that doesn't always fit.

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Of course 4 + 8 are never really  equivalent to 6 + 6 in the real world.

This is the philosophy section. What does the "real world" have to do with it?

___

4 + 8, 8 + 4, and 6 + 6 are equivalent in arithmetic. All of the other example given are not strictly arithmetic problems. You have added context to them, which is the source of any failure of equivalence. But IMO it's also due to equivocation, because you aren't using + to mean addition in its normal context anymore.

# Philosophy

Philosophy (from Greek φιλοσοφία, philosophia, literally "love of wisdom")[1][2][3][4] is the study of general and fundamental concerns such as existence, knowledge, matter, values, reason, mind, and language.[5][6]

We model the real world in abstractions and call it "philosophy" or "epistemology" but it's still the real world.   If philosophy were not representative of reality it would have no value.

Additionally, philosophy underpins our beliefs and understanding of reality.  Whether we see 4 + 8 = 6 + 6 as true or not is fundamental to both the questions posed here and philosophy.

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Posted (edited)
On 3/3/2019 at 8:32 PM, 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.

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)

If we disregard if the object either bolt or nut. correct, the answer is 12, to be specific,  12 objects all in all

But to have 4 nuts and 8 bolts, assuming both nuts and bolts are matched, only 4 are usable, then you have to return back to the store and present the store man that the 4 bolts have missing nuts,  you need 4 more  nuts. so all in all there will be 16 objects

Same thing, if there are 8 nuts and 4 bolts, you simply have to go back to the store man and say, that you have 4 nuts and need 4 more  bolts. So all in all, again there are 16 objects.

The thing is, you need 8 pairs of bolts and nuts to make them useful.

Now, if both 6 bolts and 6 nuts were bought together, at the same time,  then only 6 pairs of bolts and nuts can be useful.

Using the above statements, therefore, we have more than three outcomes of possibilities.

But as far as I know, in  Algebra x cannot be y, unless x = y.

Therefore,  6x+6y will still be 6x+6y and we write it that way (not 12 x and y), and can no longer simplify it.

Same as 4x+8y can never be 8x+4y, unless of course,  x=y

so we can say 4x+8y = 8x+4y  and even say 4x+8x = 8y+4y

Again,  6+6 = 12; 8+4 = 12; 4+8 = 12  are all correct, as a general statement

but on the other hand, if you specify  the number of oranges, nuts or bolts individually, then it will be a different story.

Somehow, 6 nuts + 6 bolts , 4 nuts + 8 bolts and 8 nuts + 4 nuts are 12 OBJECTS , to be a valid statement.

Edited by Sirjon
simply to simplify

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Posted (edited)
On 3/3/2019 at 4:00 AM, studiot said:

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

Good question!

4, 6, and 8 are natural numbers, we all agree.

Are 4, 6, and 8 integers?

Are 4, 6, and 8 rational numbers?

Are they real numbers?

Are they complex numbers?

Are they quaternions?

Then why aren't they octonions?

Discuss.

Edited by wtf

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3 hours ago, wtf said:

Then why aren't they octonions?

Is octonian arithmetic associative?

Is it distributive?

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Posted (edited)
2 hours ago, studiot said:

Is octonian arithmetic associative?

Is it distributive?

The octonions have a subfield isomorphic to the reals in which all the usual arithmetic properties apply. We identify the real numbers with that subfield and say that "the octonions contain the reals" or "the octonions contain a COPY of the reals," depending on how formal we want to be.

It's no different than asking if the integers are rational. Strictly speaking, the integers are only isomorphic to a subring of the rationals; but the identification is always made unless we are being EXTREMELY picky.

So if you deny that the octonions contain the reals, you'd also have to deny that the rationals contain the integers and that the reals contain the rationals. The identifications are standard.

It's interesting that we're in the Philosophy section. Because this is indeed a point of philosophical interest. Are the rationals a subset of the reals? Or are they only isomorphic to some subset of the reals? Should we care, or should we just make the natural identification and not worry about it? We're always told that "math is based on set theory," but that is falsified by this example. As sets, the integers are not a subset of the rationals and the rationals are not a subset of the reals. But nobody ever cares to make that picky point, which shows that even though we give lip service to founding everything on set theory, we don't always mean it literally.

Edited by wtf

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Posted (edited)
On 3/3/2019 at 5:53 AM, 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.

It can be, but 6+6 is not the same as 4+8, or 8+4.

Mathematically my mind works so long as I can remember the rules. It also helps if I understand the reasons for the rules. Actually for me that is even more important, because it helps with understanding in the future.

It is when I face math that has been turned into a conversation that I have my problems begin. Begin a math question with ( if a ) (and) (if a ) and I am completely lost almost immediately.

Now, I'm wondering what if I had actually learned how to have a conversation first? Maybe I wouldn't be so analytically slow, and it seems to me that math is analytics.

Is this what you mean by restrictive? I have noticed that people do get impatient with me when trying to teach or tell me something they become impatient which leaves me having to guess which simply makes things more difficult to understand.

I have also noticed that when I say people it is often assumed by the person I am addressing that I mean them when actually I am in effect showing that I think that 6+6 is not the same as 4+8 , and it really really does confuse me when I say people and the assumption becomes personal.

Now that I think about it my whole life has seemed somewhat restrictive. Now what do I do?

Edited by jajrussel

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Posted (edited)
1 hour ago, jajrussel said:

. Now what do I do?

Well, ya' could start inventing a cardinal mathematics where the first of six oranges and  the second of six oranges ... and the 6th of 6 oranges plus the first of six apples ... and the sixth of six apples equals(?) 12 servings of fruit.

The world is complex but mathematics only needs to be logical so long as we remember while we apply it to the real world.  "Proper" mathematics can be as complex as the real world.

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2 hours ago, jajrussel said:

It can be, but 6+6 is not the same as 4+8, or 8+4.

Mathematically my mind works so long as I can remember the rules. It also helps if I understand the reasons for the rules. Actually for me that is even more important, because it helps with understanding in the future.

It is when I face math that has been turned into a conversation that I have my problems begin. Begin a math question with ( if a ) (and) (if a ) and I am completely lost almost immediately.

Now, I'm wondering what if I had actually learned how to have a conversation first? Maybe I wouldn't be so analytically slow, and it seems to me that math is analytics.

Is this what you mean by restrictive? I have noticed that people do get impatient with me when trying to teach or tell me something they become impatient which leaves me having to guess which simply makes things more difficult to understand.

I have also noticed that when I say people it is often assumed by the person I am addressing that I mean them when actually I am in effect showing that I think that 6+6 is not the same as 4+8 , and it really really does confuse me when I say people and the assumption becomes personal.

Now that I think about it my whole life has seemed somewhat restrictive. Now what do I do?

Congratulations on being the only responder who caught the main point of my topic and didn't get side tracked.

+1

On 3/3/2019 at 10:53 AM, studiot said:

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

Apologies to evryone else for my not being clear enough.

Now if I could only remember those threads I was talking about.

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Well, ya' could start inventing a cardinal mathematics where the first of six oranges and  the second of six oranges ... and the 6th of 6 oranges plus the first of six apples ... and the sixth of six apples equals(?) 12 servings of fruit.

The world is complex but mathematics only needs to be logical so long as we remember while we apply it to the real world.  "Proper" mathematics can be as complex as the real world.

Unless I have misunderstood the OP I think it was backwards. I think that thinking 6+6 is the same as 4+8 is too restrictive by limiting a person's ability to understand the complexities if the real world, but I may have misunderstood.

Imagine my surprise to have learned that some people think a biscuit is a cookie, and my disappointment to realize I said no to a cookie. Understanding that 6+6 is not necessarily the same as 4+8 can be seen as less restrictive especially since I would have never said no to a cookie.

Hmm the OP has replied. I need to go read it.

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2 minutes ago, jajrussel said:

Unless I have misunderstood the OP I think it was backwards. I think that thinking 6+6 is the same as 4+8 is too restrictive by limiting a person's ability to understand the complexities if the real world, but I may have misunderstood.

Imagine my surprise to have learned that some people think a biscuit is a cookie, and my disappointment to realize I said no to a cookie. Understanding that 6+6 is not necessarily the same as 4+8 can be seen as less restrictive especially since I would have never said no to a cookie.

Hmm the OP has replied. I need to go read it.

...And it would have been more restrictive to have said "no" to a first cookie and a second cookie.

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...And it would have been more restrictive to have said "no" to a first cookie and a second cookie.

Hmm, seems logical maybe I need to put it to a test which is what science is about. If it turns out that that the generally accepted definition of a biscuit is a cookie. I am not going to insist that my thought is better, but will yield to the much better higher learned authorities. After all it would be foolish of me to keep insisting that I am right in such a circumstance.

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