The speed of light was defined by a meter, but then it suddenly swapped so that the meter became defined by the speed of light! I guess that equals signs go both ways, but that is THE MOST RECURSIVE mathematical phenonema to ever exist!
The nice thing about the speed of light is that it is constant. That means that if you measure the distance that light travels in a vacuum in one second and then divide it by 299,792,458, you will always get the same distance, and that distance will equal one meter.
Imagine I'm making a meter stick. How do I know how long it should be? Well, perhaps I take an existing meter stick and use that to measure out exactly one meter. We'll assume that the stick that I'm measuring from is, in fact, exactly one meter. Now, the new meter stick I just made is made out of wood. Wood is not a perfect material. It may bend over time, it may absorb moisture and grow, etc. Any material that I use, in fact, will not be perfect: its length will vary slightly over time. For many applications, this will not be a problem. However, what happens when someone else wants to make meter sticks? They need to know how long a meter is. So perhaps they come to me and purchase my finest, most accurate meter stick, and use that to make new ones. Even my most accurate meter stick will not be enough to give them the same measurement every time: their meter sticks will not always be the same meter that my sticks are. And if someone makes meter sticks from their meter sticks, the problem will only compound.
Now, depending on what the meter sticks are being used for, the amount of variation might not matter at all. However, this raises a question: what is the actual length of a meter? You see, the physicists are knocking on my door saying that they want to have the speed of light defined in meters per second, and - since the speed of light is invariant - they want it to stay the same number of meters per second. (Physicists get quite annoyed when their invariants vary.)
Now, I might just take my original meter stick and say "This meter stick is officially one meter long. One meter is defined as the length of this stick." However, there's a problem with this: my meter stick is not made out of a perfect material either, since perfect materials don't exist! That means that its length will vary, which means that the length of the meter will vary, which means that the "constant" speed of light will vary, which means that the physicists will be be annoyed.
Unlike meter sticks, however, a number that is written down will stay that number forever. And also unlike meter sticks, the speed of light will never change. Therefore, in order to have a definition of a meter that is consistent, all I need to do is determine how many meters light travels in a fixed period of time and then write that number down. Any time I want to know how long a meter is, all I have to do is repeat the experiment and divide by the number I wrote down.
TL;DR a meter has to be defined by something, so it might as well be something that we know won't change.
None of which takes away from your point that measurements are inherently arbitrary.
Measurements are Weird
in Unnecessarily Overcomplicated
A blog by Channelknight Fadran in General
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The nice thing about the speed of light is that it is constant. That means that if you measure the distance that light travels in a vacuum in one second and then divide it by 299,792,458, you will always get the same distance, and that distance will equal one meter.
Imagine I'm making a meter stick. How do I know how long it should be? Well, perhaps I take an existing meter stick and use that to measure out exactly one meter. We'll assume that the stick that I'm measuring from is, in fact, exactly one meter. Now, the new meter stick I just made is made out of wood. Wood is not a perfect material. It may bend over time, it may absorb moisture and grow, etc. Any material that I use, in fact, will not be perfect: its length will vary slightly over time. For many applications, this will not be a problem. However, what happens when someone else wants to make meter sticks? They need to know how long a meter is. So perhaps they come to me and purchase my finest, most accurate meter stick, and use that to make new ones. Even my most accurate meter stick will not be enough to give them the same measurement every time: their meter sticks will not always be the same meter that my sticks are. And if someone makes meter sticks from their meter sticks, the problem will only compound.
Now, depending on what the meter sticks are being used for, the amount of variation might not matter at all. However, this raises a question: what is the actual length of a meter? You see, the physicists are knocking on my door saying that they want to have the speed of light defined in meters per second, and - since the speed of light is invariant - they want it to stay the same number of meters per second. (Physicists get quite annoyed when their invariants vary.)
Now, I might just take my original meter stick and say "This meter stick is officially one meter long. One meter is defined as the length of this stick." However, there's a problem with this: my meter stick is not made out of a perfect material either, since perfect materials don't exist! That means that its length will vary, which means that the length of the meter will vary, which means that the "constant" speed of light will vary, which means that the physicists will be be annoyed.
Unlike meter sticks, however, a number that is written down will stay that number forever. And also unlike meter sticks, the speed of light will never change. Therefore, in order to have a definition of a meter that is consistent, all I need to do is determine how many meters light travels in a fixed period of time and then write that number down. Any time I want to know how long a meter is, all I have to do is repeat the experiment and divide by the number I wrote down.
TL;DR a meter has to be defined by something, so it might as well be something that we know won't change.
None of which takes away from your point that measurements are inherently arbitrary.