Are these things illusions?

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That is interesting. A BTW I wonder how much was lost in the Alexandrian Library.

That would be interesting. I'd love some time to browse that library...

@Grebel

Mathematics like human language for interpersonal connection are abstract constructs created by human beings to describe the patterns and connections of nature.

But all of that starts with one digit; later named prehensile thumb, the ability to manipulate objects in the environment with fine motor coordination with the other 4 digits named as fingers, in the actual painstaking development of abstract concepts as the symbols of letters together as words and numbers that identify thumbs and fingers.

A tool may be associated from nature later labeled and named as stick, and our ancestors scratching symbols in sand by copying the imaginary connections between stars in the night sky as well as copying patterns that are observed in the rest of nature. And of course that part is even possible by just using an index finger or other digit directly in the sand.

Eventually, the result of that is human abstract constructs cooperatively developed with reciprocal social communication as symbols of numbers and an alphabet like Sanskrit

A 6 or a 9 is a approximation in reflection of PHI as the natural occurring pattern in nature.

And any symbols in whole or part coming close to the shape of a circle as an artistically created number is coming closer to a reflection of PI, as a natural occurring pattern in nature.

Sacred geometry arises from nature, human imagination, reason, and artistic creation.

And of course it is fascinating how these symbols vary between western and eastern cultures.

The symbols actually reflect the way the cultures as whole restrict the views of people in experiencing and viewing nature.

Anyway, what is observed as sacred and holy is usually what is shared by the culture and or religion; and numbers
and letters are among the most sacred of ancient symbols for good reason, as much human comfort has arisen
from the earliest inventions of the sacred tools of numbers and alphabets to assist human beings in gaining subsistence.

Numbers are a reflection of nature at the core of human creation. And interestingly, people are like numbers, in macrocosm and microcosm way as above so below; inside outside; and all around; as human in many ways are the cosmos, also known as the ALL THAT IS, also known as GOD unto itself, as Carl Sagan eloquently describes the significance of the human existential much fuller reality that imagination, creativity and science allows us to greater see with the aided human eye, including the aid of artistically created numbers reflecting nature as an endless cycle of connections in both creativity and human perceived and created meaning in nature.

The bottom line is it's not just that we are connected to each other; IT (All THAT IS AKA GOD) IS all connected and greater observed reality; there are no 'REAL' illusions in life; not even in the imagination of human being before the word illusion is created; it is all a reflection of 'REAL' nature in someway; and not all people have the ability to share the reflections of nature as whole that each unique human is able to do; as humans create their perception of the Universe as they go with unique eyes; not much different than that first human being who attempts to copy nature by scratching symbols in sand in day or night.

We can credit a prehensile thumb, four fingers, and sand for all we know in life, now as written language, as consisting of alphabet letters, numbers, collective intelligence, and culture as whole.

So a grain of sand is as important as a skyscraper, angles or the imagination of angels in sky. The moral of the entire story is don't discount the significance of a grain of sand; we depend on that grain of sand to both stand on what it comprises as whole earth; as well as to name earth and imagine, reason, and create the 'number' of grains of sand as earth with the use of relative human free will, as we reflect the grand creator of Nature AKA GOD.

And no; one will never find all of this in one textbook; as creator me; creates it with relative human free will.

Relative human free will can be empirically proven through art and originality in creativity. That's becoming rarer as time goes by, as humans become more specialized and limited in mechanical cognition mind as culture dictates; but never the less, human relative free will can be proven through the empirical results of humans who truly create with originality with the caveat highlighted that IT IS ALL RELATIVE AND CONNECTED AS WE stand upon the shoulders of giants of grains of sands.

As above so below; inside outside; all around IS more than frigging Haiku; IT IS a reflection of GOD. To understand all of this is the greater possibility to live in bliss; when exercised and practiced with human relative free will; of course..

And with all due respect to anyone reading this; just because one never sees it before; does not mean it does not exist; is insane or whatever. It is in empirical measure of human relative free will in human creativity and originality.

Before humans explain to other humans what numbers are; numbers are meaningless gibberish; however that does not discount the GRAND Importance of that grain(s) of sand; that first finger(s) and or prehensile thumb; or that first human who works with painstaking human imagination, creativity, and reason; to artistically create what that number is now as symbol for patterns and connections in nature; through individual and cooperative effort with other humans and of course again that grain of sand AKA GOD, too.

A math anti-realist would argue ...

1 apple + 1 apple = 2 apples is an illusion (called "a fiction"), because the apple objects are not objectively identical. No two apples will be physically the same.

Therefore, the basics of math, and consequently, all of math, exists only in human thought.

see:
https://en.wikipedia.org/wiki/Fictionalism

I am compelled to agree with you.

ruveyn

"1 apple + 1 apple = 2 apples is an illusion"

Perhaps stated the question the way you do creates the illusion.

For early farmers "1 apple" meant 1 apple without visible major flaws and therefor "2 apples" meant 2 apples without visible major flaws. No farmer (or buyer/trader) considered any apples exactly identical. This existed in reality until a budding mathematician decided the statement could be applied to other non-physical items without all the "longhand" description.

Therefor the concept of numbering systems and their manipulation was in effect and visibly true long before someone decided to invent mathematics.

^^^
Amen with applause for common sense; Facebook Like 1; Google Plus 1; and hurray as well..