Currently the image of apples that is used to illustrate counting has two problems: The apples look too identical, so one could say it is a picture of one apple; but . There are six apples in the picture and they could be grouped by the eye in different ways, not only the intended one. The intended way of grouping the apples as (1 single apple, a pair of 2 apples, a row of 3 apples) could be highlighted by connecting them in a colored rectangular background or other helpful way.
I thought the same thing; their (apparently exact) similarity hides the issue of "differences among identical objects", which gets into some issues about the Peano axioms and the reflexive definition of equality: "For every natural number x, x = x. That is, equality is reflexive." http://en.wikipedia.org/wiki/Peano_axioms Does 1 apple = 1 apple? What if the apples are of different size? Or of different type? Then 1 apple might not equal 1 apple ... Bruce Schuman (talk) 14:28, 3 August 2013 (UTC)
I have replaced the image by a version, helpfully created by MjolnirPants, in which the apples within each row differ. I hope this deals with the identicality issue. Maproom (talk) 06:14, 13 June 2014 (UTC)
I don't know. To me, the caption says "one apple, two apples, three apples", but the picture just shows six apples, arranged in a triangle. The idea that the one, two, three are supposed to correspond to rows of the triangle is not really obvious.
Maybe if the rows were spaced farther apart? Or if we made them different things (one apple, two cats, three waterfalls)?
But I have to say I'm a little bit skeptical as to whether this notion is really well served by an image of this sort. Our readers pretty much know what it means to count; I don't know whether they really need a picture of counted things. --06:33, 13 June 2014 (UTC)
I agree that the image does nothing to help anyone understand the article – it just makes the article look prettier. But at least it is now technically correct. Maproom (talk) 08:36, 13 June 2014 (UTC)
I can make further changes to the image if anyone thinks it would be helpful. The image is being used to illustrates sets, right? I might be able to pull off some baskets to put the apples in. MjolnirPantsTell me all about it. 12:33, 13 June 2014 (UTC)
I think baskets is a good idea. Maproom (talk) 15:28, 13 June 2014 (UTC)
Is there agreement - at least in America - about whether the natural numbers include zero? (I don't care about university-level math in this context: I want to know what to tell my students so they'll "get the question right" on the high-stakes statewide tests.) --Uncle Ed (talk) 12:12, 25 April 2013 (UTC)
I don't know about America but in Australia the students get taught that natural numbers don't include zero, because whole numbers are natural numbers INCLUDING the number zero. — Preceding unsigned comment added by 184.108.40.206 (talk) 11:23, 5 February 2014 (UTC)
No test should ever rely on this; in most countries (including the US) different people use it differently. If the test in question has ever used the concept of "natural numbers" without defining them, you should raise an official complaint. 220.127.116.11 (talk) 19:09, 11 June 2014 (UTC)
Early on: 'the Peano axioms (1889) begin the natural numbers with zero'. Later: 'Peano's original formulation, the first natural number was 1'. I think the second is correct. 18.104.22.168 (talk) 20:13, 10 May 2014 (UTC)
There is no contradiction. Note the difference in tense: the formulation changed after the original formulation. —Quondum 20:50, 10 May 2014 (UTC)
Reading a little more closely, the axioms appear to be agnostic to the interpretation of the first element; it is only when one defines the operation of addition that any difference emerges. The distinction arises from the choice of definition of the addition operation. As such, it seems that this article is making untrue statements about the Peano axioms (which do not define an addition operation, only a successor function). This seems to be a significant misrepresentation; I'd appreciate comment from those more familiar with working more formally in this area. —Quondum 21:06, 10 May 2014 (UTC)
The sentence in the lede "In 1763 W. Emerson's Method of Increments contains, on page 113, the phrase 'To find the product of all natural numbers from 1 to 100 ... .'" appears irrelevant to the point being discussed. Would anyone object if I deleted it? Maproom (talk) 23:12, 30 May 2014 (UTC)
I agree that it should be deleted, for a multitude of reasons, including that it does not allow us to deduce the first natural number. The subsequent sentence, "But the Peano axioms (1889) begin the natural numbers with zero." should also be deleted, as per my argument above. —Quondum 00:33, 31 May 2014 (UTC)
Right. I have deleted both sentences. Maproom (talk) 07:54, 31 May 2014 (UTC)