Actually the MIDI number that represents a note does have a standard. Middle C is defined as #60 (261.6256 Hz in standard tuning), and other notes are defined relative to that. The problem with the octave number designation of middle C in midi devices/applications seems to occur because the way musicians usually think of numbering the octaves is based on a piano keyboard with 88 keys/semitone pitches, while MIDI has a representation of 128 semitone pitches. If you start counting octaves at C0 going up it matters where you place C0.
MIDI C0 is the normal tuning of MIDI key #0 = 8.1758 Hz. That is not a note most musicians find useful since it is below the frequency humans can hear. So MIDI is representing more than a full octave of semitones that are lower in pitch than we can hear, and thus not part of music as we know it. MIDI note #60 (middle C in music) is 5 octaves higher than MIDI note #0 and thus logically we would call that C5 of the MIDI range.
If we decide that we will number the octaves beginning with a C, then there are a couple of semitones below the lowest C on the piano. Since they are part of the octave that began at the still subsonic C (MIDI # 12= 16.3516 Hz) the lowest white key can logically be called A0. The lowest C note on a piano keyboard is C1 the beginning of the second octave with audible notes on the piano is MIDI note #24 at 32.703 Hz. If we follow that piano keyboard numbering system MIDI note #60 (261.63 Hz) is C4, the fourth C on the keyboard and the key beginning the fifth octave in which there are any notes represented.
So if you think your keyboard/sequencer is a MIDI representation you would call middle C (MIDI note # 60) C5, the note beginning the 6th octave of the MIDI range. If you want to convert it to make it more understandable as a musical instrument you would call it C4.
As Scook notes, it is a pretty trivial thing to program your software or digital synth to change that representation