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Major revision (7/06/2023): silly mistakes which confused many people have been amended, and I have better elucidated the question.

For a square, area (unit²) = side length × side length. However, this does not answer my question.

Here is what I want to draw attention to: you can interpret area iteratively by using increasingly larger amounts of increasingly smaller squares (areas) to represent an area of a given value. Thus, area seems to make sense only in the context of smaller versions of itself. In other words, using infinitely larger numbers of infinitely smaller squares to represent an area value, highlights the recursive nature of the mathematical concept of area.

If the above is accurate, and I don't want to bite off more than I can chew here, a logical consequence would be that unless area can be infinitely small, area is illogical.

However, this question seems more appropriate to the broader context of numbers, i.e., 2 contains two ones and so on.

More pertinent to the concept of area would be asking how or why does squaring a value (which is what a unit is) represent two dimensional space?

Growing6884
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    There is a history of mathematical results related to what you are asking about "area". One might begin with the notion of length and how to define it (for an interval this is easy, but what about for curves?). In general the topic [tag:measure-theory] was introduced to provide some rigor in defining area and volume, and it was discovered that some interesting problems are uncovered. A narrower Question by you would no doubt elicit some valuable content. – hardmath Jun 04 '23 at 14:44
  • It is a measure of two dimensional space. It answers the question, how much of the space do I have? Our formal definitions of concepts like length, area and volume are chosen to capture that idea. – John Douma Jun 04 '23 at 15:43
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    "What do we have 25 of?" You wrote it yourself: square meters. – Арсений Кряжев Jun 04 '23 at 16:14
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    Related (duplicate?): "Area in axiomatic geometry" – Blue Jun 04 '23 at 16:28
  • @АрсенийКряжевiswithUkraine that is correct and I will edit my question, however, it still does not answer my question which is mostly to do with the recursive nature of the concept of area. – Growing6884 Jun 05 '23 at 05:26
  • @Simon See also What is the formal, precise definition of area (and that of volume) in geometry? – dxiv Jun 05 '23 at 05:45
  • @JohnDouma I understand, albeit this does not answer my question, which may be my own fault, for not being clear enough. I would argue that it answers the question, what is the value of the sum of the areas of the square subunits? However, if we keep mathematically defining smaller and smaller subunits, the definition or 'answer' remains the same. Thus, according, to dxiv it does not answer the "undefined notion of amount of space". – Growing6884 Jun 05 '23 at 05:48
  • Also could anyone enlighten me as to why my post has been downvoted, I would genuinely like to rectify any errors I have made. – Growing6884 Jun 05 '23 at 05:51
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    @Simon I did not downvote, but it's not clear why your question is about areas, specifically, and not about lengths, too, since both are primitive concepts that can be accepted or postulated or defined in different ways at different levels. Yet, you seem to not have a problem with lengths, but just with areas. – dxiv Jun 05 '23 at 05:55
  • @dxiv thank you for asking this wonderful question. I do have a problem with the concept of length, in fact I have a problem with every concept, ever! I make that argument here: https://philosophy.stackexchange.com/questions/96658/do-distinctions-and-concepts-exist. – Growing6884 Jun 05 '23 at 06:11
  • @Simon Your question here about areas uses the concept of length a lot. It helped if you added what explanation or definition for length you rely on, or found satisfactory enough to use here. That could provide some sense of what kind of answers about areas you are looking for. – dxiv Jun 05 '23 at 06:25
  • Thanks @dxiv, I will preface by saying that I am not versed in mathematics. Nonetheless, currently, I do not see how the interpretation of length or distance would change the interpretation of the question. This is because I thought that the concept of length had only a single interpretation. However, I do understand that their is a connection between the concepts of length and area.

    If a distance measures the amount of space between two objects, then this distance between the two objects can be infinitely divided into smaller distances or spaces, which results in the same conundrum.

     – Growing6884 Jun 05 '23 at 08:24
  • I hope this helps resolve any uncertainties in my question. If you have any further recommendations, do not hesitate to let me know. I can apply your feedback to the question if need be. – Growing6884 Jun 05 '23 at 08:24
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    The answer depends on the context which is badly missing here. For instance, in Analysis one isually means "2-dimensional Lebesgue measure." In elementary geometry the answer depends on axioms that you use (one for Hilbert, another for Birkhoff). One can also say that Euclid does not have notion of area but of "equidecomposibility." – Moishe Kohan Jun 05 '23 at 11:21
  • @MoisheKohan thanks for attempting to clarify, could you elaborate for the less mathematically literate (me and others), on the relevance of context and mention what is a '2-dimensional Lebesgue measure', what are the Hilbert and Birkhoff axioms, and what is 'equidecomposibility'? If not, no problem, but maybe you could answer my question by doing so, thanks. – Growing6884 Jun 05 '23 at 12:24
  • Take a look at my answer here and see if it helps. – Moishe Kohan Jun 05 '23 at 12:53
  • The fact that it's infinitely divisible does not make the notion recursive. You can say the same thing with numbers: 1 is 100 times 1/100, which is 100 times 1/10000 and so on, ad infinitum. This is not at odds with the notion of a number, which is a thing we can manipulate in a certain way. Similarly, the area is an assignment of numbers to shapes fulfilling some properties. We agree that a square of length 1 has this thing as 1 and use its properties (which we impose) to calculate it for more complex shapes. – Арсений Кряжев Jun 05 '23 at 18:10
  • @MoisheKohanonstrike thank you I highly appreciate the help, though as a second year Biology student, who did miserably in standard-level high school mathematics I doubt I will be familiar with the specialised terminology and concepts you are using. Nonetheless, I will be self-studying maths in order to broaden my knowledge. Until then, I am restricted to 'conversationally expressed' logic. – Growing6884 Jun 06 '23 at 09:46
  • @АрсенийКряжевiswithUkraine thank you, although, I still maintain that my question has not been answered. In my mind, whereas a number represents a 'quantity', area represents a quantity of space, what is this space, exactly? – Growing6884 Jun 06 '23 at 09:58
  • @hardmath you and dxiv raise excellent points and I whole-heartedly agree, I should have 'attacked' the notion of length, as it is essentially the same problem, but without less 'distractions' (1 dimension versus two dimensions). Furthermore, I have misled people into thinking 5 m² was composed of 25 1 m² units, this is not true. A silly mistake by me that confused a lot of people. Foolishly, I thought that squaring the lengths of the square would result in the area (m²), so a square with lengths 5 × 5 would equal 5 m², rather than 25 m². – Growing6884 Jun 07 '23 at 12:59
  • Can you state clearly what you are willing to accept? For example, are you willing to accept the real number line and its properties? I ask this because one answer would be to explain that the concept of area is logically founded upon the concept of length of a segment in the real number line. But you yourself pointed out that the same "recursive" issue applies to length of a segment in the real number line, so if you are not willing to accept the concept of length in the real number line, using that to explain area would be pointless. – Lee Mosher Jun 07 '23 at 13:42
  • @LeeMosher I understand the nature of the website is to get things answered, although I do think questions can be answered for different people through different approaches, so irrespective of what I accept or not, an explanation would always benefit someone. – Growing6884 Jun 08 '23 at 07:37
  • @LeeMosher thank you, I hadn't heard of a 'real number line' until you said it. You say: "But you yourself pointed out that the same 'recursive' issue applies to length of a segment in the real number line", you essentially know my standpoint. Regardless, I am forced to 'accept' helpful concepts, which are like tools in that maybe we don't know how they work exactly, but they can achieve things. You say: "if you are not willing to accept the concept of length in the real number line, using that to explain area would be pointless", an explanation is always better than no explanation. – Growing6884 Jun 08 '23 at 09:45
  • Regarding your comment on "different approaches", I should emphasize for you that this is a mathematics web site, the goal of which is to give mathematical answers to mathematical questions. If what you want is a philosophical explanation of the implications of infinite subdivision --- which seems from your post like a distinct possibility --- then this site is not a good place for that. If on the other hand you are interested in the mathematical foundations of area, the link provided by @MoisheKohanonstrike is excellent. – Lee Mosher Jun 08 '23 at 13:28
  • Thank you @LeeMosher I will do my best to read and understand the answer provided by MoisheKohanonstrike. – Growing6884 Jun 09 '23 at 01:42

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I guess you could say that area is the amount of space a 2-dimensional shape is enclosing. A 1 m^2 square can be divided into 10000 squares with area of 1 cm^2. So, one could simply say that the amount of space enclosed by a square with an edge of 1 m is equivalent to the sum of the amount of space enclosed by 10000 squares with edge 1 cm.

Teufel
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  • I don't see how this answers OP's question. It merely replaces the undefined notion of "area" with the undefined notion of "amount of space". – dxiv Jun 05 '23 at 05:39
  • I agree and that is exactly what I have tried to communicate in my original question. Thank you for clarifying. However, this does not answer my question as @dxiv has succinctly described. – Growing6884 Jun 05 '23 at 05:39
  • @Teufel my apologies on a silly error in my original post, which misled you and others. 1 m² is composed of 100 cm² units (100 cm * 100 cm, L × W). Yet the original message still stands, you can interpret area iteratively by using increasingly larger amounts of increasingly smaller squares (areas) to represent an area of a given value. into thinking 5 m² was composed of 25 1 m² units, this is not true. A silly mistake by me that confused a lot of people. – Growing6884 Jun 07 '23 at 13:05
  • Initially and foolishly, I thought that squaring the lengths of the square would result in the area (m²), so a square with lengths 5 × 5 would equal 5 m², rather than 25 m². Whereas a square with lengths 2.5 × 2.5 would equal 5 m². What a mind bender. – Growing6884 Jun 07 '23 at 13:06