Is addition a more fundamental concept than subtraction?

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It is fairly obvious that addition is a more flexible tool than counting, but addition might be applied to achieve the same goal achieved by counting. Counting to discover a number of cows or concubines – and then boasting about the number – is an ancient pastime.

Qualitative matters – such as how you feel about your own existence – seem to go beyond what numbers can measure. Even the idea of how fundamental a concept is seems to be a qualitative matter.

Subtraction has an important application: to compute distances.

We can assign labels to points based on their distance from the reference point zero, along a straight line in a particular direction. For example, the point labelled (1/2) is half a unit from zero, and the point labelled (1/3) is one third of a unit from zero. Given that assignment of labels, provided that we move in the same direction away from zero for the two points, the value of the expression ((1/2) minus (1/3)) is equal to the distance between the point marked (1/2) and the point marked (1/3).

If we were to move in opposite directions from zero to get to our points, then the distance would be the value of the expression ((1/2) plus (1/3)). This can be explained in terms of negative numbers. We cannot use the same label (1/2) for a point left of zero and a point right of zero if we want to use the labels to compute distances.

Before Solomon was born, people knew that a distance of half a unit is quite different from half of a baby. You can cut an interval on a straight line in half without killing the interval.
 
Addition comes first. Even biologically, we seem to have a sense of relative quantification.

For subtraction, one needs to have a sense of the ‘zero’ and the negative.

ICXC NIKA
 
It’s an inverse operation. They are equal. And even though addition and subtraction have fact families, you are posting in the wrong forum…
 
Hoosier_Daddy said:
It’s an inverse operation. They are equal. And even though addition and subtraction have fact families, you are posting in the wrong forum…
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There is a forum for philosophical musings about the historical evolution of mathematics?

I think when you advance from counting to the addition operator, the subtraction operator is going to come with it. I have seven, add three, I get ten, and take the three away, I’m back down to seven. One of the imaginative leaps that underpin discussions of jet propulsion is hardly required.
 
EasterJoy said:
the historical evolution of mathematics?
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Let’s not use the “e” word (evolution). Biological evolution happens to be a forbidden topic on Catholic Answers Forums.

I suspect that participants in this thread look inside their own minds, influenced by one small fragment of the past: how they learned the ideas. Now, trying to sort things out and remove that personal bias, a judgment is reached.

Two different judgments have already been expressed:
GEddie said:
Addition comes first.
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I take that to be a “Yes” response to the thread title question: “Is addition more fundamental than subtraction?”
Hoosier_Daddy said:
It’s an inverse operation. They are equal.
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I take that to be a very specific “No” answer, expressing a belief that addition and subtraction are equally fundamental.

Although two mutually exclusive views have been expressed, I hope that the debate can remain polite, unlike past debates on Catholic Answers Forums about the “e” topic.
 
PseuTonym said:
Let’s not use the “e” word (evolution). Biological evolution happens to be a forbidden topic on Catholic Answers Forums.

I suspect that participants in this thread look inside their own minds, influenced by one small fragment of the past: how they learned the ideas. Now, trying to sort things out and remove that personal bias, a judgment is reached.

Two different judgments have already been expressed:

I take that to be a “Yes” response to the thread title question: “Is addition more fundamental than subtraction?”

I take that to be a very specific “No” answer, expressing a belief that addition and subtraction are equally fundamental.

Although two mutually exclusive views have been expressed, I hope that the debate can remain polite, unlike past debates on Catholic Answers Forums about the “e” topic.
Click to expand...
I do not understand this post.
 
Hoosier_Daddy said:
I do not understand this post.
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I don’t know where to begin to provide a remedy if the only symptom is not understanding. Perhaps it is analogous to “there is a discomfort or pain or problem in my body somewhere.” A doctor would likely ask you to point. The body has many parts.

Question #1 regarding description of symptoms:
Re-reading from the beginning of the problematic message, what is the first sentence that you consider unclear?

Question #2 regarding description of symptoms:
Looking back on the whole message, what one sentence is especially lacking in clarity?
 
PseuTonym said:
Let’s not use the “e” word (evolution). Biological evolution happens to be a forbidden topic on Catholic Answers Forums.

I suspect that participants in this thread look inside their own minds, influenced by one small fragment of the past: how they learned the ideas. Now, trying to sort things out and remove that personal bias, a judgment is reached.

Two different judgments have already been expressed:

I take that to be a “Yes” response to the thread title question: “Is addition more fundamental than subtraction?”

I take that to be a very specific “No” answer, expressing a belief that addition and subtraction are equally fundamental.

Although two mutually exclusive views have been expressed, I hope that the debate can remain polite, unlike past debates on Catholic Answers Forums about the “e” topic.
Click to expand...
The evolution of philosophy is by no means forbidden on Catholic Answers. The word evolution in its broader sense merely refers to the gradual development of something, such as from a simple to a more complex form. The opposite would be something like regression or reversion.

You can think what you like, but I can hardly imagine that someone who has the insight to realize without counting that when they have five objects and obtain seven objects more they will always have twelve objects will not likewise see that if they have twelve and lose five they will be left with seven. Why? Because the first persons to have the insight that lead to the concept of addition–the insight that they could know the result of combining two known quantities without re-counting everything–did not learn this by rote.

It is only the person who has learned 7+5 = 12 as a rote “math fact” who will not see that the operation of addition is readily reversible.
 
EasterJoy said:
It is only the person who has learned 7+5 = 12 as a rote “math fact” who will not see that the operation of addition is readily reversible.
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I don’t know how general a claim you are making when you say that “the operation of addition is readily reversible”, but I do believe that there can be a very big gap between an individual, fully-confirmed computational result, and a general principle.

That somebody doesn’t accept a general principle that you believe to be true doesn’t necessarily imply that the person learned by rote. Sometimes what is thought to be a reliable principle is discovered to be false. Before the discovery is made, people might believe in the principle, such as because they are misled by their experience with examples of the principle being true, and their lack of experience with counter-examples, or lack of imagination in constructing counter-examples.
 
EasterJoy said:
It is only the person who has learned 7+5 = 12 as a rote “math fact” who will not see that the operation of addition is readily reversible.
Click to expand...
I have two questions that you should take at face value. I’m not intending to be argumentative, but simply want more information about what you mean by “math fact” and “rote.”

#1 Is it a “math fact” that the negative numbers are left of the zero point on the x-axis, and that the positive numbers are right of the zero point on the x-axis? From the point of view of some high school teachers, it is a fact. However, from my point of view, a labelled x-axis displays a function (call it “f”) mapping points to numbers. There happens to be a perfectly good function g having the property, for every x in its domain, g(x) = (- x). The composition h of the functions f and g having the property, for every point p on the x-axis, h(p) = g(f(p)) is then associated with an “incorrect” labeling of the x-axis. However, from my point of view, a construct cannot be “incorrect”, and the non-existence of composition functions would create serious difficulties.

#2 Do students learn by rote that, for all real numbers x, y, and z, (x + y) + z = x + (y + z)?
 
PseuTonym said:
I have two questions that you should take at face value. I’m not intending to be argumentative, but simply want more information about what you mean by “math fact” and “rote.”

#1 Is it a “math fact” that the negative numbers are left of the zero point on the x-axis, and that the positive numbers are right of the zero point on the x-axis? From the point of view of some high school teachers, it is a fact. However, from my point of view, a labelled x-axis displays a function (call it “f”) mapping points to numbers. There happens to be a perfectly good function g having the property, for every x in its domain, g(x) = (- x). The composition h of the functions f and g having the property, for every point p on the x-axis, h(p) = g(f(p)) is then associated with an “incorrect” labeling of the x-axis. However, from my point of view, a construct cannot be “incorrect”, and the non-existence of composition functions would create serious difficulties.

#2 Do students learn by rote that, for all real numbers x, y, and z, (x + y) + z = x + (y + z)?
Click to expand...
No. By a “math fact,” I mean the stuff you memorize in grade school: addition tables, multiplication tables. Some students do not immediately see the connection between addition and subtraction, but those aren’t students with the insight that would have made the leap between reaching a sum by recounting a combined total all over again and seeing immediately that some certain sum and some other sum always combine to give a definite third sum. Once you’ve made that realization yourself, rather than simply being assured by someone “good at math” that the relationship exits, I believe you must also recognize that the operation is reversible.

No, deciding how to express the situation where you have a debt that exceeds the sum you have on hand is another different leap than realizing that you can learn to predict the results when you add or take away pre-counted amounts to counted quantities without counting the whole new collection all over again. At an early stage, if you said, “What is the result when you take twelve sheep away from a herd of five,” of course it is reasonable to say, “Who on earth thinks you can go into a sheep pen with five sheep and lead twelve sheep out? It can’t be done. There is no way you can take five away from twelve.” Likewise, there will be some at the beginning who think it is pretty silly to talk about leading no sheep at all out, as if you’ve done something or to believe there is some mathematical calculation involved in asking how many sheep you have when you start with five and remove the five. On a concrete level: nothing is nothing. You don’t count it. You don’t add it, you don’t subtract it. Why would you need a number for it? It is nothing.

The leap to seeing the usefulness of a numeral to express zero, of negative numbers, and even of using a “base” or bundling to conceptualize and express numbers, those are different leaps. I could be persuaded to believe they could be arrived at in a different order, excepting that I don’t think negative numbers make much sense without the concept of zero.

You could conceivably add, subtract, multiply, and divide and use negative numbers without using a base or “bundling” system, although a base is so convenient of course I’d predict that mathematics would be so cumbersome that might few would enjoy it much before someone make the leap between naming acceptable “bundles” of objects to the elegant solution of a base numerical system. (Roman mathematics must have been a royal pain in the hind end before they were tipped off to the more elegant Arab methods.)
 
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