OK, looked at the pictures. To be honest I still cannot say for sure. If I had to guess I would say it looks like water spots or something on the outside of the rear element, I just cannot tell. If you have a small flashlight try shining it through from the front and see if you can determine which lens element and which side of the element the spots are on.
Do you have eyeglass cleaner available? And a microfiber cloth? terrycloth will do if you are careful, but never use paper. Try gently wiping the rear element with the cleaner, just a little at a time to see if the spots can be removed.
If they are on the outside then you should be able to clean them, another thought would be to take the lens to your local eyeglass store, and smile nice to the lady and she might clean it for you.
The bad news is that if the spots are on the inside of the rear element it is a long way in to get to it. Takumars are built from the back forward, to open them you take off the front element and disassemble toward the back. In this case the rear element is the last thing you take out. You have to go through the aperture mechanism to get to it. I have never gone that deep, the lenses I've repaired had issues on either the front or second elements.
Here is what the elements of that lens will look like inside, front element is on the left.
Fungus spores are in the air all the time, and present inside all lenses. They just do not usually grow. If the right conditions are present then it can grow almost over night. It needs warmth, moisture and food to grow just like everything else. Usually what happens is the lens get wet, and gets some water inside. The fungus starts to grow but quickly runs out of either food or water and dies. If that is fungus it probably grew and died years ago. If it was still growing the whole lens would be covered, just like a slice of bread that gets mold, its the same thing almost.