For what reason are data sets with no repeating values defined as no mode whilst sets with n/2 is part of the set of integers is defined? This question is in regards to the reasoning underpinning the definition of a set with no repeating values as having no mode.  I am interested in the analytical philosophy behind this.
A set with no repeating values is defined as having no mode. But a set that has an even number of elements is described as having a median = [(n/2)th term + (n/2)+1 term)]/2 rather than “no median”.
So, in cases where there is no repeating element, for what reason did mathematicians not define such sets as mode = mean?
 A: A mode is, by definition, the value most represented.  If no value is represented more than any other, then there is no mode.  (Which possible value could it be?)
The case is different for median, which is the value for which half of the data is above, and half of the data is below.  Now there always DOES exist such a value (for $n>1$).  True, there may be multiple such values, and here it is just a useful and principled convention to choose the average value of all those consistent with the constraint.
A: Here is a different perspective on finding
the 'mode' of a sample from a continuous
distribution (where there are theoretically
no ties.)
Sometimes one tries to use a large sample from a continuous
population distribution to estimate the mode of the population.
Consider the following sample of size $n = 5000$ from
a population distributed $\mathsf{Beta}(3, 5),$
which has mode $2/6 = 0.3333$ according to this Wikipedia page. Of course, in a real application we would not know the population mode. [Using R.]
set.seed(225)
x = rbeta(5000, 3, 5)
summary(x)
    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
0.01159 0.25535 0.36605 0.37609 0.48496 0.91409 

Using a histogram. Here is a histogram of the data:
    hdr = "n=5000: From BETA(3,5)"
    hist(x, prob=T, col="skyblue2", main = hdr)


The 'modal interval' of this histogram has the tallest
bar with base $(0.30, 0.35),$ so one might guess that
the mode of the population is between $0.30$ and $0.35.)$
Some elementary statistics books have formulas for
guessing the mode, supposedly more precisely, by
considering heights of nearby histogram bars.
Bear in mind that various histograms (with different bins) might result in somewhat different answers.
Using a density estimator. Independently of how any particular histogram of the data may be drawn, a kernel density estimator (KDE) attempts
to imitate the PDF of the population from the available data.
Below we show the KDE for our data (brown curve), and also the PDF
of the population distribution (dotted curve, which would ordinarily not be known in a real application).
lines(density(x), col="brown", lwd=2)
curve(dbeta(x, 3,5), add=T, lwd=2, lty="dotted")


The KDE in R is a sequence of 512 points $(x,y).$
So we can look for the highest point on the KDE plot, which corresponds to about $0.31$ on the horizontal axis.
xx = density(x)$x
yy = density(x)$y
mode = xx[yy==max(yy)];  mode
[1] 0.3080652

So, according to the KDE, the mode of the population
is estimated as about $0.31.$
