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When we look at the moon, we see a spherical object, but do "spheres" really exist? This may seem to be a silly question, because it's not hard to understand the definition of a sphere: "the set of all points in space that are a fixed distance (called the radius) from a fixed point (called the center)." We see examples of "spherical" objects all the time, don't we?
23 First, nothing that we can observe in our physical world 24 complies perfectly to this mathematical definition of a sphere. The moon, a beach ball, and even water droplets are all "bumpy," at least at the atomic level. So can we say that the concept of "sphere" is real 25 if there is no such thing as a real sphere?
Pondering this question as so many ancient Greek philosophers did, 26 the argument Plato made was that the sphere is an "ideal form," inaccessible to our physical senses yet 27 the mind can apprehend it through pure reason.
He also reasoned that, since our senses can be fooled, logic provides a much more reliable path to the truth. Therefore, a Platonic idealist believes that these abstract forms are 28 as effective, if not more so, than sensory experience at revealing the nature of reality. 29
Modern scientists and philosophers are unlikely to be Platonic idealists. Today, we can understand the origin of abstract concepts 30 and not having to believe that they come from a higher, physically inaccessible reality. We simply need to understand 31 the process by which our brains make inferences.
Take an abstract idea like "orangeness." Most of us would say that orangeness "exists" because we see examples of it every day, such as carrots, traffic cones, and pumpkins. But what if, by some magic, we could remove all orange-colored objects from the universe? In other words, what if, as with "sphereness," no real examples of "orangeness" 32 would exist? Would "orangeness" still exist?
In an important sense, the answer is yes. We can demonstrate the existence of "orangeness" without appealing to any higher reality. We could measure the wavelength of red light (about 650 nm), and yellow light (about 570 nm) and make the reasonable inference, because wavelengths fall on a continuum, that a color exists with an intermediate wavelength, of 610 nm, even if we have never directly measured such light.
Our brains do not contain sophisticated instruments for measuring wavelengths of light, but they do make similar inferences constantly. 33 For instance, when you drive, you unconsciously make inferences about quantities like the speeds of surrounding cars and qualities like dangerous driving conditions. Our brains are continually making inferences based on the limited information from our senses, and these inferences are the substance of abstract thought.
29. At this point, the author is considering adding the following true statement:
The sphere is just one of many ideal forms, like lines and tetrahedrons, that are studied in geometry.
Should the author make this addition here?
33. Which of the following changes would best improve this sentence's cohesiveness with the rest of the paragraph?
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