I am amused, on the fret board, why when you go up and down, does the same note name change to a sharp, going up of the note before and a flat going down of the note before, meaning that a an E flat is the same sound as a D sharp. Why is it so?
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That seems like an innocent enough question on the surface… but in order to really understand the reasons behind the “system” that we have, you need to dive quite deeply into music history and potentially physics and even politics as well.
I would suggest you start out looking up the terms “equal temperament” and “enharmonic” and see from there how deep you want to go 
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There are a few different reasons. The easiest way to think about it is that sharp and flat are directional.
Sharp means go higher in pitch. Flat means go lower in pitch.
Alot of this goes back to how music is written. When a key is written out each letter only appears once. Meaning, in a given key you can’t have both an A and an A#. It would show as an A and a B flat. It’s all part of the complexity of communicating music that has been influenced by hundreds of years of music theory.
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Well, yeah, maybe I misunderstood the question… but there really is no way around a bit of a deep dive if this is even to make a bit of sense 
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Yep. Other than the very basics, I still can’t communicate effectively about music theory.
Unfortunately, this is true. To understand why music theory is the way it is… I’m guessing that’s master’s degree level of study.
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You are asking a grammar question.
It would be like asking why, in English, does the word “wood”, (as in a tree or stick or board) sound the exact same as “would”, (as in the conditional, past tense form of will).
They mean different things, they are spelled differently, but they sound the same.
It is the same with two notes with different names and spellings that sound the same. (These are called enharmonics). The reason they have the different options for how they are described and written has to do with musical grammar and context.
Certain keys use different notes.
Your ears will never know the difference.
As @joergkutter and @eric.kiser put it above, the technical reasons behind which musical grammar rules to use when (why D# and not Eb?) require a fair amount of music theory knowledge to provide enough context.
Let me know if you want that deep dive.
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It’s about the scale in the key that the song is in. Some will be flat some will be sharp. If you follow the key the note indicator would reveal itself. By the way same confusing numbers also apply to the note/ keys. Like 9, 11, 13. It sounds crazy but once you get to work on it it’s just pretty simple.
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Homophones?
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Aye, see what you did.
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Think of a sharp as adding a 1/2 and a flat as subtracting a half… so you can have 2+1/2 or 3-1/2.
BUT whether you’re going up or down has no effect on whether a note is sharp or flat; that depends on the key of the scale eg the key of G has one sharp: F#. It’s F# going up, it’s F# going down, it does not have a Gb.
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Thank you all, complex to say the least, I love the grammar analogy, thank you all for responding, I will be happy just to think of wood and would and keep playing …
Norm 
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The actual reason is historical: There are a dozen ways to tune an instrument. They have names like “just” and “tempered” etc. In each method of tuning, how a note sounds is slightly different. Only in one of these tunings, called “Equal Temperament”, the note values are defined so that the values of D# and Eb overlap and end up as the same note. We call this “Enharmonic Confusion” in German, which shows that originally this was an unusual thing. It just happens that by now this system is the one used by most people, because it’s a good compromise between the dis-/advantages of all others.
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Bit off-topic but this topic reminded me of Igor Stravinsky who wrote a piece of music that caused a riot. It is said people couldn’t mentally process the music and dance and got violent. Great story 
Reverb made a short video about it:
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Thanks Paul, great thought process
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Igor Stravinsky, progenitor of the mosh pit.
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