First, is minimalize a word?
If so, what is the difference between the two words minimalize and minimize, and when should each be used?
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Several dictionaries do not have an entry for minimalize (such as Oxford Dictionaries and Merriam-Webster). Still, other dictionaries do (such as Random House, via Dictionary.com and Webster's New World and American Heritage, via YourDictionary), and the word is definitely in use, so the answer to the first part of your question is yes, it's a word.
The difference between minimize and minimalize is very subtle. In many cases, either word would work and they would mean the same thing. However, some would argue that minimalize implies striving for minimalism, which isn't quite the same as simply making as small as possible. One blogger who describes himself as a minimalist specifically uses both words in the title of an article, in an effort to highlight the difference: "The Goal is to Minimalize Consumption, not to Minimize It".
Some claim that minimalize means something akin to "portray as small or unimportant"; whereas minimize means to actually make something small. This distinction doesn't jibe with the definitions for minimize given in various dictionaries covering both of those uses. For example, from Oxford Dictionaries:
Reduce (something, especially something unwanted or unpleasant) to the smallest possible amount or degree: the aim is to minimize costs
Represent or estimate at less than the true value or importance: they may minimize, or even overlook, the importance of such beliefs
I believe that this misconception comes from conflating minimalize with marginalize:
[Again, from Oxford Dictionaries]:
Treat (a person, group, or concept) as insignificant or peripheral: attempting to marginalize those who disagree
Agreed. They don't mean exactly the same:
According to Macmillan:
to reduce something harmful or unpleasant to the smallest amount or degree: We must minimize the damage to innocent civilians.
to make something seem much less important than it really is: "I don't want to minimize their role in the campaign.
"minimalize" would suggest "to reduce something as much as possible, as minimalists tend to do"